On sets minimizing their weighted length in uniformly convex separable Banach spaces
aa r X i v : . [ m a t h . F A ] N ov ON SETS MINIMIZING THEIR WEIGHTED LENGTH INUNIFORMLY CONVEX SEPARABLE BANACH SPACES
THIERRY DE PAUW, ANTOINE LEMENANT, AND VINCENT MILLOT
Abstract.
We study existence and partial regularity relative to the weightedSteiner problem in Banach spaces. We show C regularity almost everywherefor almost minimizing sets in uniformly rotund Banach spaces whose modulusof uniform convexity verifies a Dini growth condition. Contents
1. Introduction 12. Preliminaries 43. Existence 74. Almost minimizing sets in arbitrary Banach spaces 135. The excess of length of a nonstraight path and a regularity theorem 206. Application to the quasihyperbolic distance 317. Differentiability of almost minimizing curves in 2 dimensional rotundspaces 32References 381.
Introduction
This paper contributes to the study of one dimensional geometric variationalproblems in an ambient Banach space X . We address both existence and partialregularity issues. The paradigmatic weighted Steiner problem is( P ) ( minimize R C w d H among compact connected sets C ⊆ X containing F .
Here H denotes the one dimensional Hausdorff measure (relative to the metricof X ), w : X → (0 , + ∞ ] is a weight, and F is a finite set implementing the boundarycondition.Assuming that problem ( P ) admits finite energy competing sets, we prove ex-istence of a minimizer in case X is the dual of a separable Banach space, and w is weakly* lower semicontinuous and bounded away from zero, Theorem 3.6. Ideason how to circumvent the lack of compactness that ensues from X being possiblyinfinite dimensional go back to M. Gromov, [9], and have been implemented byL. Ambrosio and B. Kirchheim [2] in the context of metric currents, as well as byL. Ambrosio and P. Tilli [3] in the context of the Steiner problem (with w ≡ w ; the relevant lower semicontinuityof the weighted length is in Theorem 3.4.In studying the regularity of a minimizer C of problem ( P ), we regard C asa member of the larger class of almost minimizing sets. Our definition is less This work has been partially supported by the Agence Nationale de la Recherche, through theproject ANR-12-BS01-0014-01 GEOMETRYA and ANR 10-JCJC 0106 AMAM.. restrictive than that of F.J. Almgren [1] who first introduced the concept. A gaugeis a nondecreasing function ξ : R + \ { } → R + such that ξ (0+) = 0. We saya compact connected set C ⊆ X of finite length is ( ξ, r ) almost minimizing inan open set U ⊆ X whenever the following holds: For every x ∈ C ∩ U , every0 < r r such that B ( x, r ) ⊆ U , and every compact connected C ′ ⊆ X with C \ B ( x, r ) = C ′ \ B ( x, r )one has H ( C ∩ B ( x, r )) (1 + ξ ( r )) H ( C ′ ∩ B ( x, r )) . One easily checks that if C is a minimizer of ( P ) then it is ( ξ, ∞ ) almost minimizingin U = X \ F , where ξ is (related to) the oscillation of the weight w , Theorem 3.8.For instance if w is H¨older continuous of exponent α then ξ ( r ) behaves asymptoti-cally like r α near r = 0.In order to appreciate the hypotheses of our regularity results, we now makeelementary observations. In case card F = 2 and w is bounded from above andfrom below by positive constants, each minimizer C of ( P ) is a minimizing geodesiccurve Γ with respect to the conformal metric induced by w , with endpoints thoseof F , Theorem 3.9. Since H (Γ) < ∞ we infer that Γ is a Lipschitz curve. Ingeneral not much more regularity seems to ensue from the minimizing property ofΓ. Indeed in the plane X = ℓ ∞ with w ≡
1, every 1-Lipschitz graph over one ofthe coordinate axes is length minimizing, as the reader will happily check. Howeverif X is a rotund Banach space, then Γ must be a straight line segment. Finally,in case w is merely H¨older continuous the Euler-Lagrange equation for geodesicscannot be written in the classical or even weak sense, and our regularity resultsdo not seem to entail from ODE or PDE arguments, even when the ambient space X = ℓ is the Euclidean plane.In Section 4 we report on some properties of ( ξ, r ) almost minimizing sets C ingeneral Banach spaces X . It is convenient – but not always necessary – to assumethat the gauge ξ verifies a Dini growth condition, specifically that ζ ( r ) = Z r ξ ( ρ ) ρ d L ( ρ ) < ∞ , for each r >
0. We show that for each x ∈ C ∩ U the weighted density ratioexp[ ζ ( r )] H ( C ∩ B ( x, r ))2 r is a nondecreasing function of 0 < r min { r , dist( x, X \ U ) } , Theorem 4.7. Itslimit as r ↓
0, denoted Θ ( H C, x ), verifies the following dichotomy:
Either Θ ( H C, x ) = 1 Or Θ ( H C, x ) > / , Corollary 4.8. We then establish that reg( C ) := C ∩ U ∩{ x : Θ ( H C, x ) = 1 } isrelatively open in C ∩ U and that for each x ∈ reg( C ) and every δ > < r < δ such that C ∩ B ( x, r ) is a Lipschitz curve that intersects bdry B ( x, r )exactly in its two endpoints, Theorems 4.9 and 4.11.In Section 5 we improve on the regularity of reg( C ). Assume the ambient Banachspace X is uniformly rotund and let δ X ( ε ) denote its modulus of uniform rotundity,see 2.10. Let x ∈ reg( C ). We assume r j ↓ C ∩ B ( x, r j ) is a Lipschitzcurve Γ j with endpoints x − j and x + j on bdry B ( x, r j ). We let L j be the affine linecontaining x − j and x + j . We want to show that Γ j does not wander too far awayfrom L j , i.e. we seek for an upper bound of max z ∈ Γ j dist( z, L j ). Suppose thismaximum equals h j r j and is achieved at z ∈ Γ j . The triangle inequality implies or strictly convex or uniformly convex EIGHTED LENGTH 3 H (Γ j ) > k z − x − j k + k x + j − z k . As X is uniformly rotund, the latter is quantitativelylarger than the length of the straight line segment joining x − j and x + j , specifically k z − x − j k + k x + j − z k > k x + j − x − j k (1 + δ X ( Ch j )) , Theorem 5.2. On the other hand, the almost minimizing property of C says that H (Γ j ) (1 + ξ ( r j )) k x + j − x − j k . It now becomes clear that h j cannot be too large, in fact h j C ( δ − X ◦ ξ )( r j ) , which in turns yields the Hausdorff distance estimatedist H (Γ j , L j ∩ B ( x, r j )) C ′ ( δ − X ◦ ξ )( r j ) . Upon noticing that the good radii r j can be chosen in near geometric progression,we infer that the sequence of affine secant lines { L j } is Cauchy provided ∞ X j =1 ( δ − X ◦ ξ )(2 − j ) < ∞ . (1)The fact that the relevant inequalities are also locally uniform in x then yields ourmain C regularity Theorem 5.5 under the assumption that δ X and ξ verify theDini growth condition (1).In case ξ ( r ) ∼ = r α , a change of variable shows that (1) in fact involves solely δ X ,namely it is equivalent to asking that ∞ X j =1 δ − X (2 − j ) < ∞ . (2)The condition is met for instance by all L p spaces, 1 < p < ∞ , as shown by theClarkson inequalities. The case when X = ℓ n is a finite dimensional Euclideanspace has been worked out for instance in [14] (see also [5, Section 12] and [13]).In Section 6 we apply our existence and regularity results to quasihyperbolicgeodesics for instance in L p spaces. It is perhaps worth noticing that even in thefinite dimensional setting X = ℓ np , 2 < p < ∞ , the problem is not “elliptic”, orrather the metric is not Finslerian, as the smooth unit sphere S ℓ np has vanishingcurvature at ± e , . . . , ± e n . In fact, in case X is finite dimensional and the unitsphere S X is C ∞ smooth, (2) may be understood as a condition on the order ofvanishing of f v : T v S X → R : h
7→ k v + h k − ,v ∈ S X . With this in mind, we show in Section 7 how to completely dispensewith (2) in case dim X = 2, and the norm of X is rotund and C . The relevantregularity result Theorem 7.7 states that reg( C ) is made of differentiable curves (notnecessarily C ) provided C is ( ξ, r ) almost minimizing and √ ξ is Dini. In order toprove this we localize the modulus of continuity δ X ( v ; ε ) relative to each direction v ∈ S X . We then consider the subset G = S X ∩ { v : ∂ h,h f v (0) > } . We observeit is relatively open in S X , and its complement S X \ G is nowhere dense becausethe norm is rotund, i.e. the unit circle S X contains no line segment. Furthermore,if v ∈ G then δ X ( v ; ε ) > c ( v ) ε , the best case scenario for regularity. To prove thedifferentiability of reg( C ) at x ∈ reg( C ) we need only to establish that the set oftangent lines Tan( C, x ) is a singleton. This set is connected, according to D. Preiss,[15]. Thus either L ∈ Tan(
C, x ) ∩ G = ∅ and we can run the regularity proof ofSection 5 “in a cone about L ”, or Tan( C, x ) ⊆ S X \ G and therefore Tan( C, x ) is asingleton.
THIERRY DE PAUW, ANTOINE LEMENANT, AND VINCENT MILLOT Preliminaries . —
In a metric space (
E, d ) we define the open and closed r -neighborhoods of a subset A ⊆ E by the relations U ( A, r ) = E ∩ { y : dist( y, A ) < r } B ( A, r ) = E ∩ { y : dist( y, A ) r } where dist( y, A ) = inf { d ( y, x ) : x ∈ A } . If A = { x } is a singleton, these are theusual open and closed balls U ( x, r ) and B ( x, r ). The interior, closure and boundaryof a subset S ⊆ E are respectively denoted by int S , clos S and bdry S .2.2 (The ambient Banach space X ) . — Throughout this paper X denotes a Banachspace with dim X >
2. We do not merely care about the isomorphic type of X ,but also about the specific given norm. Changing the norm for an equivalent oneaffects the corresponding Hausdorff measure, and therefore also the solutions ofthe variational problems we are interested in, as well as their regularity theory.Various collections of further requirements about X are made in distinct sections.Specifically:(3) In Section 3, X is the dual of a separable space;(4) In Section 4, X is an arbitrary separable Banach space;(5) In section 5, X is uniformly rotund, and the main result 5.5 applies when δ − X verifies a Dini growth condition, δ X being the modulus of uniformrotundity of X ;(6) In section 6, X is as in section 5;(7) In section 7, X is a finite dimensional (uniformly) rotund space with C smooth norm, and the main result 7.7 also assumes that dim X = 2.2.3 (Hausdorff distance) . — In a metric space (
E, d ) we define the
Hausdorffdistance between two closed sets A , A ⊆ E asdist H ( A , A ) = inf { r > A ⊆ B ( A , r ) and A ⊆ B ( A , r ) } . If (
E, d ) is compact then the
Blaschke selection principle asserts that ( K ( E ) , dist H )is a compact metric space, where K ( E ) denotes the collection of nonempty compactsubsets of E . It is easily seen that(I) If lim n dist H ( A n , A ) = 0 and x ∈ A then there is a sequence { x n } in E such that x n ∈ A n , n = 1 , , . . . , and lim n d ( x n , x ) = 0;(II) If lim n dist H ( A n , A ) = 0, x ∈ E and { x n } is a sequence in E such that x n ∈ A n , n = 1 , , . . . , and lim n d ( x n , x ) = 0, then x ∈ A .2.4 (Hausdorff measure) . — Given a metric space (
E, d ) we will consider the H defined for subsets A ⊆ E by thefollowing formulas: H δ ) ( A ) = inf (cid:26) X i ∈ I diam A i : { A i } i ∈ I is a finite or countable family ofsubsets of E such that A ⊆ ∪ i ∈ I A i and diam A i δ for all i ∈ I (cid:27) corresponding to each 0 < δ ∞ , and H ( A ) = sup δ H δ ) ( A ) . All Borel subsets of E are H measurable in the sense of Caratheodory, and thedefinition of H ( A ) remains unchanged if we restrict to closed (resp. open ) covers { A i } i ∈ I in the definition of H δ ) ( A ). If we want to insist about the underlyingmetric space we will write H E instead of H . It is useful to note that if F ⊆ E is EIGHTED LENGTH 5 considered as a metric space (
F, d ↾ F × F ) then H E ( F ) = H F ( F ). Finally, if E isa normed linear space and a, b ∈ E , we define the line segment with endpoints a, b by [[[ a, b ]]] = E ∩ { a + t ( b − a ) : 0 t } , and one checks that H ([[[ a, b ]]]) = k b − a k .2.5 (A covering theorem) . — Given a metric space (
E, d ) and C ⊆ E , we definethe enlargement of C asˆ C = B ( C, C )) = E ∩ { x : dist( x, C ) C ) } . In particular \ B ( x, r ) ⊆ B ( x, r ), x ∈ E , r > Vitali cover of A ⊆ E is a collection C of closed subsets of E with the followingproperty: For every x ∈ A and every δ > there exists C ∈ C such that x ∈ C and diam C < δ . It follows from [8, 2.8.6] that C admits a disjointed subcollection C ∗ with the following property: For every finite F ⊆ C ∗ one has A \ ∪ F ⊆ ∪{ ˆ C : C ∈ C ∗ \ F } . . — Given a metric space (
E, d ) and a finite Borelmeasure µ on E , we define at each x ∈ E the following generalized upper density: e Θ ( µ, x ) = lim δ → + sup (cid:26) µ ( C )diam C : x ∈ C ⊆ E, C is closed, and 0 < diam C < δ (cid:27) . If A ⊆ E is Borel, < t < ∞ , and e Θ ( µ, x ) > t for every x ∈ A , then µ ( A ) > t H ( A ) . In order to prove this we fix δ > ε >
0, and we choose an open set U ⊆ E containing A such that µ ( U ) ε + µ ( A ). Our assumption guarantees that witheach x ∈ A and i ∈ { , , . . . } large enough we can associate a closed set C x,i ⊆ U such that x ∈ C x,i , 0 < diam C x,i < i − δ , and µ ( C x,i ) > t (1 − ε )(diam C x,i ). Weextract a disjointed subfamily { C j } j ∈ J of { C x,i : x ∈ A, i ∈ { , , . . . } , C x,i ⊆ U } according to 2.5. Since for each finite F ⊆ J one has X j ∈ F diam c C j X j ∈ F diam C j − ε ) − t − µ ( E ) < ∞ , and since diam C j > j ∈ J , we infer that J is at most countable. Thuswe as well assume J = N and we may select k large enough for ∞ X j = k +1 diam c C j ε . Thus, H δ ) ( A ) k X j =1 diam C j + ∞ X j = k +1 diam c C j (1 − ε ) − t − k X j =1 µ ( C j ) + ε (1 − ε ) − t − ( ε + µ ( A )) + ε . Letting ε → δ → . — Here we consider a complete separable metric space (
E, d ) and a Borel function f : E → R . We recallthat f ( A ) is L measurable whenever A ⊆ E is Borel, see e.g. [4, Lemma 8.6.1and Corollary 8.4.3]. It thus follows as in [8, 2.10.10] that the multiplicity function R → N ∪ {∞} : r card( A ∩ f − { r } )is L measurable.Assuming furthermore that f be Lipschitz, the Eilenberg’s inequality [8, 2.10.25]states that Z R card( A ∩ f − { r } ) d L ( r ) (Lip f ) H ( A ) . THIERRY DE PAUW, ANTOINE LEMENANT, AND VINCENT MILLOT
We will often apply these two results to the case when f ( x ) = d ( x, x ), x ∈ X .2.8 (Curves) . — A curve in a metric space ( E, d ) is a topological line segment,i.e. a set Γ ⊆ E of the type Γ = γ ([ a, b ]) where a < b are real numbers and γ : [ a, b ] → E is an injective continuous map. We call γ ( a ) and γ ( b ) the endpoints of Γ, and we write ˚Γ := Γ \ { γ ( a ) , γ ( b ) } . If x ∈ Γ is not an endpoint then Γ \ { x } hastwo components. If H (Γ) < ∞ then there exists an injective γ ′ : [0 , H (Γ)] → E such that Lip γ ′ γ ′ = Γ as well. If S ⊆ E is compact connected, and H ( S ) < ∞ , then for each distinct x, x ′ ∈ S there exists a curve contained in S whose endpoints are x and x ′ (see e.g. [3, 4.4.7]).2.9 (Gauges and Dini Gauges) . — Given an interval I = R ∩ { r : 0 < r b } ,0 < b < ∞ , a gauge on I is a nondecreasing function ξ : I → R +such that lim r → ξ ( r ) = 0. We often omit to specify the interval I when it isclearly determined by the context. We say that a gauge ξ on I is a Dini gauge provided ζ ( r ) := Z r ξ ( ρ ) ρ d L ( ρ ) < ∞ ,r ∈ I , and we call ζ the mean slope of ξ . Notice ζ is a gauge as well.The following are useful examples of gauges. If ξ ( r ) ar α , a >
0, 0 < α
1, wecall ξ a geometric gauge and we easily check that it is Dini with ζ ( r ) = α − ξ ( r ). Asanother class of examples we consider the gauges ξ ( r ) = a | log r | − − α , 0 < r < α > a >
0. We call these log-geometric gauges and we checkthey are Dini as well, with ζ ( r ) = α − a | log r | − α . The gauge ξ ( r ) = | log r | − ,0 < r <
1, however, is not
Dini.Let β >
1. Define I j = [ β − ( j +1) , β − j ], j ∈ N . For any gauge ξ in I and any I j ⊆ I one has (cid:18) β − β (cid:19) ξ ( β − ( j +1) ) L ( I j ) (cid:18) inf ρ ∈ I j ξ ( ρ ) ρ (cid:19) Z I j ξ ( ρ ) ρ d L ( ρ ) L ( I j ) sup ρ ∈ I j ξ ( ρ ) ρ ! ( β − ξ ( β − j ) . Thus the appropriate comparison tests imply that ξ is Dini if and only if ∞ X j =0 I j ⊆ I ξ ( β − j ) < ∞ . Furthermore, ∞ X j = k ξ ( β − j ) (cid:18) ββ − (cid:19) ζ ( β − ( k − ) , whenever k is sufficiently large for I k − ⊆ I . Given r such that β r ∈ I andchoosing j such that r ∈ I j , this also implies that ξ ( r ) ξ ( β − j ) (cid:18) ββ − (cid:19) ζ ( β − ( j − ) (cid:18) ββ − (cid:19) ζ ( β r ) . . — We recall that a Banach space X is called uniformly rotund (abbreviated UR ) whenever the following holds. For every ε > or uniformly convex EIGHTED LENGTH 7 there exists δ > x, y ∈ B X , k x − y k > ε ⇒ (cid:13)(cid:13)(cid:13)(cid:13) x + y (cid:13)(cid:13)(cid:13)(cid:13) − δ . (3)Notice that, corresponding to a fixed ε >
0, the set of those 0 < δ < ε δ ( ε ) forwhich (3) is valid. In fact, given an arbitrary Banach space X and 0 < ε
2, weput δ X ( ε ) = inf (cid:26) − (cid:13)(cid:13)(cid:13)(cid:13) x + y (cid:13)(cid:13)(cid:13)(cid:13) : x, y ∈ X, max {k x k , k y k} k x − y k > ε (cid:27) . It is most obvious that δ X is a gauge. One notices that X is uniformly rotund ifand only if δ X ( ε ) > < ε
2. In this case δ X is called the modulus ofuniform rotundity of X .We also define δ − X ( t ) = sup { ε > δ X ( ε ) t } and we readily infer that δ X ( ε ) t implies ε δ − X ( t ) for all ε > t >
0. Thegauge δ − X , particularly its growth, pertains to the regularity theory of Section 5.2.11. Remark. —
In the definition of δ X ( ε ) one may require that k x k = k y k = 1instead of max {k x k , k y k}
1. This leads to an equivalent definition of rotundity. Existence . —
In this section X isthe dual of a separable Banach space, with norm k · k . Its closed unit ball B X equipped with the restriction of the weak* topology of X is a compact separatedtopological space. It is metrizable as well, owing to the separability of a predualof X . We let d ∗ denote any metric on B X compatible with its weak* topology, forinstance d ∗ ( x , x ) = X n − n |h y n , x − x i| , where y , y , . . . is a dense sequence of the unit ball of some predual of X . Noticethat d ∗ ( x , x ) k x − x k . In the compact metric space ( B X , d ∗ ) we denote thecorresponding Hausdorff distance as dist ∗ H . We consider two metrizable topologieson B X : that induced by the norm of X , and that induced by the weak* topologyof X . When we refer to closed (resp. compact ) subsets C ⊆ B X we always meanstrongly closed (resp. compact), i.e. with respect to the norm topology of X , andwe use the terminology weakly* closed (resp. weakly* compact ) otherwise.3.2. Lemma. —
Let ( E, d ) be a metric space. (A) If C ⊆ E is connected, x ∈ C and < r diam C , it follows that H ( C ∩ B ( x, r )) > r ;(B) If Γ if a curve in E with endpoints a and b then H (Γ) > d ( a, b ) .Proof. (A) There is no restriction to assume C is nonempty. Given x ∈ C weconsider the Lipschitz function u : E → R : y d ( y, x ). Since C is connected so is u ( C ), and r ∈ clos u ( C ) whenever 0 < r diam C . As Lip u r = H R ([0 , r [) = H R ( u ( B ( x, r ) ∩ C )) H E ( B ( x, r ) ∩ C ) . (B) follows from (A) on letting C = Γ, x = a , and r = d ( a, b ). (cid:3) Definition 1.e.1 (Vol. II Chap. 1 Paragraph e) in [11].
THIERRY DE PAUW, ANTOINE LEMENANT, AND VINCENT MILLOT
Lemma. —
Every sequence { C n } of nonempty compact subsets of B X containsa subsequence { C k ( n ) } such that dist ∗ H ( C k ( n ) , C ) → as n → ∞ for some nonemptyclosed set C ⊆ B X .Proof. Upon noticing that each C n is weakly* compact, this becomes a consequenceof the Blaschke selection principle applied to the compact metric space ( B X , d ∗ ),and the fact that a weakly* compact set C is closed. (cid:3) Theorem (Compactness and lower semicontinuity) . —
Assume that (A) { C n } is a sequence of nonempty compact connected subsets of B X ; (B) dist ∗ H ( C n , C ) → for some nonempty closed subset C of B X ; (C) w : B X → (0 , + ∞ ] is weakly* lower semicontinuous and sup n Z C n w d H < ∞ . It follows that (D) C is compact and connected; (E) R C w d H lim inf n R C n w d H ; (F) F ⊆ C whenever F ⊆ C n for every n = 1 , , . . . . Remark. —
If the function w fails to be weakly* lower semicontinuous, con-clusion (E) does not need to hold, as the following counterexample shows. Denoteby { e k } ∞ k =1 the canonical orthonormal basis of X = ℓ , and define w : X → [1 ,
2] by w ( x ) := max { , − x, span { e } ) } . Then consider the sequence { C n } ⊆ B X of compact connected sets C n := γ n ([0 , γ n ( t ) := te n for 0 t / , e n + ( t − ) e for 1 / < t / , (1 − t ) e n + e for 7 / < t . One easily checks that 3.4 (B) holds with C = [0 , e ]. On the other hand we have R C n w d H = for every n = 1 , , . . . , while R C w d H = > . Proof.
Conclusion (F) is a trivial consequence of assumption (B). If inf n diam C n =0 then diam ∗ C = lim n diam ∗ C n lim inf n diam C n = 0, thus C is a singleton andthere is nothing to prove. We henceforth assume that a := inf n diam C n > B X together with the nonvanishing and weak* lowersemicontinuity of w guarantee that η := inf B X w >
0. Therefore H ( C n ) η − R C n w d H , n = 1 , , . . . , and it ensues from (C) that b := sup n H ( C n ) < ∞ .We claim that the sequence of metric spaces { C n } is equicompact. Indeed given r > n = 1 , , . . . , and x , . . . , x κ n in C n which are pairwise a distance at least 2 r apart, it follows from Lemma 3.2 that κ n r κ n X k =1 H ( C n ∩ B ( x k , r )) H ( C n ) b , whence κ n is bounded independently of n . It follows from the Gromov compactnessTheorem (see e.g. [3, 4.5.7]) that there exists a compact metric space ( Z, d Z ), asubsequence of { C n } which we still denote as { C n } , and isometric embeddings i n : ( C n , k · k ) → ( Z, d Z ) such that D n := i n ( C n ) converge in Hausdorff distance in Z to some compact set D ⊆ Z .We now consider the mappings j n := i − n : ( D n , d Z ) → ( C n , k ·k ). We claim that,restricting to a subsequence of { D n } if necessary (still denoted by { D n } ), thereexists a 1-Lipschitz map j : ( D, d Z ) → ( B X , k · k ) with the following property: Forany sequence { z k ( n ) } in Z satisfying z k ( n ) ∈ D k ( n ) , n = 1 , , . . . , and z k ( n ) → z ∈ D , EIGHTED LENGTH 9 we have d ∗ ( j k ( n ) ( z k ( n ) ) , j ( z )) → . In order to prove this we consider the graphsof j n , G n = ( Z × B X ) ∩ { ( z, j n ( z )) : z ∈ D n } . According to the Blaschke selection principle { G n } subconverges in Hausdorff dis-tance, in the compact metric space ( Z, d Z ) × ( B X , d ∗ ), to some compact set G .One readily checks that the projection of G on Z equals D . In addition, we observethat for any pair ( z , x ) , ( z , x ) ∈ G we can find ( z n , j n ( z n )) , ( z n , j n ( z n )) ∈ G n such that { ( z nk , j n ( z nk )) } n converges to ( z k , x k ) in ( Z, d Z ) × ( B X , d w ), k = 1 ,
2, as n → ∞ . Referring to the weak* lower semicontinuity of k · k , and to the fact that j n is an isometry, we infer that k x − x k lim inf n →∞ k j n ( z n ) − j n ( z n ) k = lim inf n →∞ d Z ( z n , z n ) = d Z ( z , z ) . Consequently G is the graph in Z × B X of a 1-Lipschitz map j : ( D, d Z ) → ( B X , k·k ),i.e. G = { ( z, j ( z )) : z ∈ D ) } . In order to complete the proof of our claim, we need toestablish the asserted property of j . We consider a sequence { z k ( n ) } in Z such that z k ( n ) ∈ D k ( n ) , n = 1 , , . . . , and d Z ( z k ( n ) , z ) → z ∈ D . Any subsequenceof { j k ( n ) ( z k ( n ) ) } contains a subsequence itself converging weakly* to some x ∈ B X .The Hausdorff distance convergence of { G n } to G then implies that ( z, x ) ∈ G ,i.e. x = j ( z ). Since this is independent of the original subsequence, the conclusionfollows.We now establish that C = j ( D ), starting with the inclusion C ⊆ j ( D ). Given x ∈ C we choose x n ∈ C n , n = 1 , , . . . , such that d ∗ ( x n , x ) →
0. Letting z n := i n ( x n ), n = 1 , , . . . we infer from the compactness of Z that a suitable subsequence { z k ( n ) } of { z n } converges to some z ∈ Z . The Hausdorff convergence of { D n } to D implies that z ∈ D , and in turn the claim of the preceding paragraph impliesthat lim n d ∗ ( x k ( n ) , j ( z )) = lim n d ∗ ( j k ( n ) ( z k ( n ) ) , j ( z )) = 0, thus x = j ( z ). The otherway round, given z ∈ D we choose z n ∈ D n , n = 1 , , . . . , such that d Z ( z n , z ) → d ∗ ( j n ( z n ) , j ( z )) →
0. Since j n ( z n ) ∈ C n we conclude that j ( z ) ∈ C .The connectedness of D follows from that of each D n , the Hausdorff convergenceof { D n } to D and the relation β := sup n H ( D n ) < ∞ , in the following fashion.Given z , z ∈ D and n = 1 , , . . . , we choose a curve Γ n ⊆ D n with endpoints z , z . Since H (Γ n ) β we may select a parametrization γ n : [0 , → Z of Γ n sothat Lip γ n β . It follows from the Arzela-Ascoli Theorem and the compactness of Z that some subsequence of { γ n } converges uniformly to some Lipschitz γ : [0 , → Z . One readily checks that Γ = im γ is a curve in D with endpoints z and z .Conclusion (D) follows at once from the equality C = j ( D ).We now turn to proving conclusion (E). There is no restriction to assume thatlim inf n R C n w d H < ∞ and, extracting a subsequence of { D n } in the first place,we may also assume that this limit inferior is a limit:lim inf n Z C n w d H = lim n Z C n w d H . We also notice that there is no restriction to assume diam
D >
0, for if diam D = 0then H ( C ) H Z ( D ) = 0, because Lip j
1, and (E) is trivially verified.With each n = 1 , , . . . we associate a finite Borel measure µ n on Z by theformula µ n ( B ) = Z B ∩ D n w ( j n ( z )) d H Z ( z ) ,B ⊆ Z Borel. Since j n is an isometry we observe that µ n ( Z ) = Z D n w ( j n ( z )) d H Z ( z ) = Z C n w ( x ) d H ( x ) . (4) Thus { µ n } is bounded in C ( Z ) ∗ and it follows from the Banach-Alaoglu and Riesz-Markov Theorems that some subsequence, still denoted { µ n } , converges weakly* in C ( Z ) ∗ to a finite Borel measure µ . We establish now that e Θ ( µ, z ) > w ( j ( z )) (5)for every z ∈ D .Fix z ∈ D , 0 < r < diam D and ε >
0. Choose z n ∈ D n , n = 1 , , . . . , so that d Z ( z n , z ) →
0. Choose next z ′ n ∈ D n \ B ( z, r ) and a curve Γ n ⊆ D n with endpoints z n and z ′ n , according to 2.8. If n is sufficiently large then B ( z n , r/ ⊆ B ( z, r ),and arguing as in Lemma 3.2 we infer the existence of ˜ z n ∈ Γ n ∩ bdry B ( z n , r/ in , i = 1 ,
2, denote the two components of Γ n \ { ˜ z n } . Upon noticing thatdiam Γ in > r/ i = 1 ,
2, we infer from Lemma 3.2 that H Z (Γ in ∩ B (˜ z n , r/ > r/ i = 1 ,
2, and therefore H Z ( D n ∩ B (˜ z n , r/ > r . (6)Considering a subsequence if necessary we may assume that d Z (˜ z n , ˜ z ) → z ∈ Z . Now we abbreviate ρ = r/ ε where ε > − ε ) ρ r/
3, and we further consider only integers n so large that d Z ( z n , z ) < ε and d Z (˜ z n , ˜ z ) < ε . One then readily checks that z ∈ B (˜ z, ρ ) and that B (˜ z n , r/ ⊆ B (˜ z, ρ ). It follows from the latter and (6) that(1 − ε )(diam B (˜ z, ρ )) (1 − ε )2 ρ r H Z ( D n ∩ B (˜ z n , r/ H Z ( D n ∩ B (˜ z, ρ )) . (7)We now make the additional assumptions that w be Lipschitz (with respect to thenorm k · k of B X ) and we observe that for every ζ ∈ B (˜ z, ρ ) one has w ( j n ( ζ )) > w ( j n ( z n )) − (Lip w ) k j n ( ζ ) − j n ( z n ) k = w ( j n ( z n )) − (Lip w ) d Z ( ζ, z n ) > w ( j n ( z n )) − w ) ρ . It follows from (7) and the above that µ n ( B (˜ z, ρ )) = Z B (˜ z,ρ ) w ( j n ( ζ )) d H Z ( ζ ) > (cid:18) inf ζ ∈ B (˜ z,ρ ) w ( j n ( ζ )) (cid:19) H Z ( D n ∩ B (˜ z, ρ )) > (cid:0) w ( j n ( z n )) − w ) ρ (cid:1) (1 − ε )(diam B (˜ z, ρ )) , for n sufficiently large. Letting n → ∞ in the above and referring to Portmanteau’sTheorem, the weak* lower semicontinuity of w , and lim n d ∗ ( j n ( z n ) , j ( z )) = 0, weinfer that µ ( B (˜ z, ρ )) > lim sup n µ n ( B (˜ z, ρ )) > (cid:0) w ( j ( z )) − w ) ρ (cid:1) (1 − ε )(diam B (˜ z, ρ )) . Letting ε → ρ r → z ∈ D we infer from 2.6 that for every 0 < t < k ∈ Z , µ ( D k ) > t k H Z ( D k ) > t Z D k e Θ ( µ, z ) d H Z ( z )where D k = D ∩ { z : t k − > e Θ ( µ, z ) > t k } . EIGHTED LENGTH 11
Since we have not discussed the measurability of e Θ ( µ, · ) we refer to [8, 2.4.10 and2.4.3(2)] for the next estimate. Summing over k ∈ Z and letting t → − yields µ ( Z ) > µ ( D ) > Z ∗ D e Θ ( µ, z ) d H Z ( z ) > Z D w ( j ( z )) d H Z ( z ) . Next we infer from the surjectivity of j and the inequality Lip j H C j ∗ ( H Z D ). Thus Z C w d H Z C w d (cid:2) j ∗ ( H Z D ) (cid:3) = Z D ( w ◦ j ) d H Z . It then follows from (4) that Z C w d H Z D ( w ◦ j ) d H Z µ ( Z ) = lim n µ n ( Z )= lim n Z D n ( w ◦ j n ) d H Z = lim n Z C n w d H . This completes the proof in case w is Lipschitz. It thus remains only to removethat assumption. To this end we introduce the Yosida approximations w k of w ,defined by the relation w k ( x ) = inf { w ( y ) + k k y − x k : y ∈ B X } ,x ∈ B X , k = 1 , , . . . . We easily check that the sequence { w k } is nondecreasing andconverges everywhere to w , and that each w k is both Lipschitz and weakly* lowersemicontinuous. Therefore, Z C w k d H lim inf n Z C n w k d H lim inf n Z C n w d H , for each k = 1 , , . . . , and the conclusion follows from the Monotone ConvergenceTheorem. (cid:3) We now consider a nonempty finite set F ⊆ X and a weakly* lower semicontin-uous function w : X → (0 , + ∞ ]such that inf X w >
0. We let C F denote the collection of connected compact sets C ⊆ X such that F ⊆ C . With each C ∈ C F we associate the weighted length L w ( C ) = Z C w d H . We consider the variational problem( P F,w ) ( minimize L w ( C )among C ∈ C F , assuming that inf( P F,w ) < ∞ . Note that this finiteness assumption holds forinstance if w is bounded on the convex hull K of F . Indeed if F = { x , x , . . . , x κ } we let C = ∪ κk =1 [[[ x , x k ]]], so that C ∈ C F and L w ( C ) (sup K w ) P κk =1 k x k − x k .3.6. Theorem (Existence) . —
Whenever F and w are as above, the variationalproblem ( P F,w ) admits at least one solution.Proof. We apply the direct method of calculus of variations. Define β := inf( P F,w )and let { C n } be a minimizing sequence such that L w ( C n ) β , n = 1 , , . . . .Given n , let x ∈ C n be such that k x − x k = max y ∈ C n k y − x k , where x ∈ F . LetΓ be a curve in C n with endpoints x and x . It follows that1 + β > L w ( C n ) > Z Γ n w d H > (inf X w ) H (Γ n ) > (inf X w ) k x − x k . Therefore C n ⊆ B ( x , R ) where R = (1 + β )(inf X w ) − and the conclusion followsfrom Theorem 3.4 applied with B ( x , R ). (cid:3) We end this section by showing that the minimizers of problem ( P F,w ) are almost minimizing in a sense to be defined momentarily, and the remaining partof the paper will be devoted to studying the regularity properties of these (moregeneral) almost minimizing sets.3.7 (Almost minimizing sets) . —
Given a gauge ξ , an open set U ⊆ X , and r > C ⊆ X is ( ξ, r ) almost minimizing in U provided H ( C ) < ∞ and the following holds: For every x ∈ C ∩ U , every 0 < r r suchthat B ( x, r ) ⊆ U , and every compact connected set C ′ ⊆ X with C ′ \ B ( x, r ) = C \ B ( x, r )one has H ( C ∩ B ( x, r )) (1 + ξ ( r )) H ( C ′ ∩ B ( x, r )) . (8)A set C ′ as above is called a competitor for C in the ball B ( x, r ).Given an open set U ⊆ X , a function w : U → R , and r >
0, we recall that the oscillation of w at scale r > w, r ) = sup {| w ( x ) − w ( x ) | : x , x ∈ U and k x − x k r } . Thus lim r → + osc( w, r ) = 0 if and only if w is uniformly continuous.3.8. Theorem. —
Assume that F ⊆ X is a nonempty finite set, that w : X → [ a, b ] (where < a < b < ∞ ) is uniformly continuous, and that the variational problem ( P F,w ) admits a minimizer C . It follows that C is ( ξ, ∞ ) almost minimizing in X \ F , relative to the gauge ξ ( r ) = osc( w, r ) (cid:18) a + ba (cid:19) . Proof.
Notice that ξ is indeed a gauge since w is both bounded and uniformlycontinuous. Define U = X \ F , and fix x and r such that x ∈ C and B ( x, r ) ⊆ U .We abbreviate B = B ( x, r ) and we observe that for each competitor C ′ in B onehas a H ( C ∩ B ) Z C ∩ B w ( x ) d H ( x ) Z C ′ ∩ B w ( x ) d H ( x ) b H ( C ′ ∩ B ) , as well as( w ( x ) − osc( w, r )) H ( C ∩ B ) Z C ∩ B w ( x ) d H ( x ) Z C ′ ∩ B w ( x ) d H ( x ) ( w ( x ) + osc( w, r )) H ( C ′ ∩ B ) . Therefore, w ( x ) H ( C ∩ B ) ( w ( x ) + osc( w, r )) H ( C ′ ∩ B ) + osc( w, r ) H ( C ∩ B ) ( w ( x ) + osc( w, r )) H ( C ′ ∩ B ) + osc( w, r ) ba H ( C ′ ∩ B ) , and the conclusion follows upon dividing by w ( x ) > a > (cid:3) Since we are considering 1 dimensional geometric variational problems, it is worthpointing out the easy local topological regularity of minimizers.
EIGHTED LENGTH 13
Theorem. —
Assume that card F = 2 and that w : X → R + \{ } is uniformlycontinuous. It follows that every minimizer of problem ( P F,w ) is a curve Γ withendpoints those of F , and that Θ ( H Γ , x ) = 1 for each x ∈ ˚Γ .Proof. If C is a minimizer then it contains a curve Γ with endpoints those of F ,according to 2.8. It follows that L w ( C \ Γ) = 0, and in turn H ( C \ Γ) = 0. Fromthis we infer that in fact C \ Γ = ∅ , for if x ∈ C \ Γ and r > B ( x, r ) ∩ Γ = ∅ then 0 < H ( C ∩ B ( x, r )) H ( C \ Γ), according to 3.2, a contradiction.Let x ∈ ˚Γ and r > B ( x , r ) ∩ F = ∅ . We choose an arclengthparametrization γ : [ a, b ] → X of Γ, and a < t < b such that x = γ ( t ). Define t − := inf { t t : γ ( t ) ∈ bdry B ( x , r ) } ,t + := sup { t > t : γ ( t ) ∈ bdry B ( x , r ) } . We create a competitor for the problem ( P F,w ) as follows: C ′ = γ ([ a, t − ]) ∪ [[[ γ ( t − ) , x ]]] ∪ [[[ x , γ ( t + )]]] ∪ γ ([ t + , b ]) . From the relation L w ( C ) L w ( C ′ ) we obtain Z γ ([ t − ,t + ]) w d H Z [[[ γ ( t − ) ,x ]]] ∪ [[[ x ,γ ( t + )]]] w d H . It entails from the definition of t + and t − that Γ ∩ B ( x , r ) ⊆ γ ([ t − , t + ])), thus infact Z Γ ∩ B ( x ,r ) w d H Z [[[ γ ( t − ) ,x ]]] ∪ [[[ x ,γ ( t + )]]] w d H . We next infer from the uniform continuity of w that( w ( x ) − osc( w ; r )) H (Γ ∩ B ( x , r )) Z Γ ∩ B ( x ,r ) w d H , as well as Z [[[ γ ( t − ) ,x ]]] ∪ [[[ x ,γ ( t + )]]] w d H ( w ( x ) + osc( w ; r ))2 r . Therefore, if r > H (Γ ∩ B ( x , r ))2 r w ( x ) + osc( w ; r ) w ( x ) − osc( w ; r ) . Letting r → + we infer that Θ ∗ ( H Γ , x )
1. The reverse inequalityΘ ∗ ( H Γ , x ) > (cid:3) Almost minimizing sets in arbitrary Banach spaces
We establish the basic discrepancy between regular and singular points of almostminimizing sets.4.1 (Local hypothesis about the ambient Banach space) . —
In this section X denotes a separable Banach space.4.2. Scholium. —
We will repeatedly use (without mention) the following ob-servation. If B ⊆ X is a closed ball of radius r > X withendpoints a and b so that a B and b ∈ B , then Γ ∩ bdry B = ∅ . This is because if γ : [0 , → X parametrizes Γ so that f (0) = a and f (1) = b , and if x is the centerof the ball B , then f ( t ) = k γ ( t ) − x k is continuous and f (1) r < f (0). In fact,there is the smallest parameter t ∗ such that γ ( t ∗ ) ∈ bdry B . Thus the subcurve Γ ′ of Γ with endpoints a and γ ( t ∗ ) is so that ˚Γ ′ ∩ B = ∅ .4.3. Proposition. —
Assume that: (A) C ⊆ X is compact and connected, H ( C ) < ∞ , x ∈ C , r > , and B = B ( x, r ) ; (B) N ∈ N \ { } , card C ∩ bdry B = N , and C ∩ bdry B = { x , . . . , x N } . If x ∈ B then C ′ = ( C \ B ) ∪ (cid:0) ∪ Nn =1 [[[ x , x n ]]] (cid:1) is a competitor for C in B . In particular, if N = 1 then C ′ = C \ int B is a competitor for C in B .Proof. Since C ′ is the union of C \ int B and finitely line segments, it is compact.We now show that any pair of a, b ∈ C ′ is connected by a curve contained in C ′ .If a and b both belong to B , they are related connected by a curve in C ′ ∩ B . Wenow assume one or both of a and b does not belong to B . Recalling 2.8, we select acurve Γ ⊆ C with endpoints a and b . If ˚Γ ∩ B = ∅ we are done. Assume a B andchoose n ∈ { , . . . , N } such that x n is closest to a along Γ, and denote Γ a,x n thecorresponding subcurve of Γ, so that Γ a,x n ⊆ C ′ . If b ∈ B then b can be joindedto x n in C ′ ∩ B by a curve Γ b,x n , and Γ a,x n ∪ Γ x n ,b ⊆ C ′ is a curve with endpoints a and b . If instead b B then let x m , m ∈ { , . . . , N } be closest to b along Γ,and denote Γ b,x m the corresponding subcurve of Γ, so that Γ b,x m ⊆ C ′ . Finally,choose a curve Γ x n ,x m contained in C ′ ∩ B with endpoints x n and x m and noticethat Γ a,x n ∪ Γ x n ,x m ∪ Γ x m ,b ⊆ C ′ is a curve with endpoints a and b . (cid:3) Theorem. —
Assume that: (A) C ⊆ X is compact and connected, U ⊆ X is open, < r < r , x ∈ C , B ( x, r ) ⊆ U , ξ is a gauge; (B) C is ( ξ, r ) almost minimizing in U ; (C) r < min(diam C, r ) .The following hold: (D) card( C ∩ bdry B ( x, r )) > ; (E) If card( C ∩ bdry B ( x, r )) = 2 then C ∩ B ( x, r ) contains a Lipschitz curvewhose endpoints are { x , x } = C ∩ bdry B ( x, r ) ; (F) If card( C ∩ bdry B ( x, r )) = 2 and r is a point of L approximate continuityof ρ card( C ∩ bdry B ( x, ρ )) then for every < ε < there exists (1 − ε ) r <ρ < r such that the following dichotomy holds: either C ∩ B ( x, ρ ) is a Lipschitz curve and C ∩ bdry B ( x, ρ ) consists of itsendpoints; or C ∩ B ( x, r ) contains three Lipschitz curves Γ , Γ , Γ whose intersectionis a singleton { ˜ x } = Γ ∩ Γ ∩ Γ , and ˜ x ∈ int B ( x, r ) is an endpointof each Γ , Γ and Γ . Remark. —
Several comments are in order.(A) The “temporary” conclusion (E) does not assert that C ∩ B ( x, r ) is a Lip-schitz curve Γ, but merely that it contains such Γ whose endpoints are onbdry B ( x, r ); in particular it is not claimed that x ∈ Γ.(B) The first alternative of conclusion (F), however, states that C ∩ B ( x, ρ ) is a Lipschitz curve Γ, and that C ∩ bdry B ( x, ρ ) consists of the endpoints ofΓ, for ρ close to r .(C) The function ρ card( C ∩ B ( x, ρ )) is L measurable on R + , recall 2.7,and hence approximately continuous L almost everywhere, see [8, 2.9.12and 2.9.13]. EIGHTED LENGTH 15
Proof of Theorem 4.4.
We abbreviate B = B ( x, r ) and we start by proving (D).Since x ∈ C ∩ B and r < diam C we infer that C ∩ bdry B is not empty. Assuming C ∩ bdry B is a singleton, we infer from 4.3 (applied with x = x ) that C ′ = C \ B is a competitor for C in B . Now since C is almost minimizing we would have H ( C ∩ B ) (1 + ξ ( r )) H ( C ′ ∩ B ) = 0, in contradiction with 3.2.We now turn to proving (E). Let C ∩ bdry B = { x , x } . We will show that C contains a Lipschitz curve Γ with endpoints x and x , whose interior ˚Γ =Γ \ { x , x } is contained in int B . Recalling 2.8 we infer that there exists a Lipschitzcurve Γ ⊆ C with endpoints x and x . If ˚Γ ∩ int B = ∅ then ˚Γ ⊆ int B for otherwisecard( C ∩ bdry B ) >
3, a contradiction. Thus (E) will be established if we rule outthe case ˚Γ ⊆ X \ B . b B = B ( x, r ) xx x b C Γ b Figure 1.
A case to rule out in proving (E).Assuming if possible that ˚Γ ⊆ X \ B we verify that C ′ = ( C \ B ) ∪ { x , x } is a competitor. It is indeed easy to check that C ′ is compact and we now showthat it is connected. Given a, b ∈ C ′ we will find a curve Γ ′ ⊆ C ′ with endpoints a and b . According to 2.8, there exists a curve Γ ′′ ⊆ C with endpoints a and b .If Γ ′′ ∩ int B = ∅ we let Γ ′ = Γ ′′ and we are done. Otherwise Γ ′′ contains oneof x and x , and hence also both. We denote by Γ ′′ the subcurve of Γ ′′ withendpoints a and (say) x , by Γ ′′ the subcurve of Γ ′′ with endpoints x and b , andwe put Γ ′′ B = Γ ′′ ∩ B . Thus Γ ′′ = Γ ′′ ∪ Γ ′′ B ∪ Γ ′′ and we define a new curve Γ ′ ⊆ C ′ corresponding to Γ ′′ ∪ Γ ∪ Γ ′′ . This completes the proof that C ′ is a competitor. Nowsince H ( C ′ ∩ B ) = 0 and C is almost minimizing, we infer that H ( C ∩ B ) = 0.Together with hypothesis (C), this contradicts Lemma 3.2. Thus conclusion (E) isestablished.It remains to prove (F). Let B r = B ( x, r ), { x , x } = C ∩ bdry B ( x, r ), andlet Γ ⊆ C denote a Lipschitz curve with endpoints x and x , and ˚Γ ⊆ int B r ,whose existence results from conclusion (E). The L approximate continuity of ρ card( C ∩ bdry B ( x, ρ )) at ρ = r implies the existence of an increasing sequence { ρ k } with limit r and such that card( C ∩ bdry B ( x, ρ k )) = 2 for every k . Choose ρ = ρ k with k large enough for (1 − ε ) r < ρ < r . Taking k even larger we mayassume that card(Γ ∩ B ( x, ρ )) > ⊆ int B r and x , x ∈ bdry B r are theendpoints of Γ. The curve Γ being a subset of C we must have card(Γ ∩ B ( x, ρ )) = 2.Abbreviate B ρ = B ( x, ρ ). If C ∩ B ρ = Γ ∩ B ρ then the first branch of the dichotomyoccurs and the proof is finished. Otherwise let { y , y } = C ∩ bdry B ρ = Γ ∩ bdry B ρ .Choose y ∈ ( C ∩ B ρ ) \ Γ. Thus C contains a curve Γ ′ with endpoints y and y . Let y ′ be the first point on Γ ′ (starting from y ) that belongs to Γ. Clearly y ′ ∈ B ρ . Whether y ′ ∈ int B ρ or y ′ ∈ { y , y } , one checks that C contains three nontrivialLipschitz curves Γ , Γ , Γ whose intersection is { y ′ } , two of which are subcurvesof Γ, the other one being a subcurve of Γ ′ . (cid:3) We now state the basic discrepancy regarding the density of points of almostminimizing sets.4.6.
Theorem. —
Assume that (A) C ⊆ X is compact and connected, U ⊆ X is open, ξ is a gauge, r > , x ∈ C ∩ U ; (B) C is ( ξ, r ) almost minimizing in U .The following hold. (C) Θ ∗ ( H C, x ) > ; (D) One of the following occurs: either ap lim r → + H ( C ∩ B ( x, r ))2 r = 1 , or Θ ∗ ( H C, x ) > / . Proof.
Given r > ρ > C ∩ bdry B ( x, ρ )) >
2. Thus2 r Z r card( C ∩ bdry B ( x, ρ )) d L ( ρ ) H ( C ∩ B ( x, r ))according to Eilenberg’s inequality, recall 2.7. Conclusion (C) readily follows.In view of (C), conclusion (D) will be established as soon as we show that thealternative holds with the first condition replaced by the formally weakerap lim sup r → + H ( C ∩ B ( x, r ))2 r . (9)We define an L measurable set G = R ∩ { r > C ∩ bdry B ( x, r )) = 2 } , and ϑ ( r ) = r − L ( G ∩ [0 , r ]). We choose 0 < r ′ r small enough for B ( x, r ′ ) ⊆ U and ξ ( r ′ ) < /
4. We claim that if r ∈ G ∩ [0 , r ′ ] then H ( C ∩ B ( x, r ))2 r ξ ( r ) (10)and ϑ ( r ) > − ξ ( r ) . (11)In order to prove (10) we recall that our assumption card( C ∩ bdry B ( x, r )) = 2implies C ′ = ( C \ B ( x, r )) ∪ ([[[ x , x ]]]] ∪ [[[ x, x ]]]) (where { x , x } = C ∩ bdry B ( x, r )) isa competitor, according to 4.3. The desired inequality thus ensues from the almostminimizing property of C . In order to establish (11) we refer to Theorem 4.4(D),to Eilenberg’s inequality 2.7, and to (10):2 L ( G ∩ [0 , r ]) + 3 L ([0 , r ] \ G ) Z r card( C ∩ bdry B ( x, ρ )) d L ( ρ ) H ( C ∩ B ( x, r )) (1 + ξ ( r ))2 r . In other words, 2 rϑ ( r ) + 3 r (1 − ϑ ( r )) (1 + ξ ( r ))2 r , from which (11) readily follows. Still assuming that r ∈ G ∩ [0 , r ′ ], the bound ξ ( r ) < / r ∈ G with r/ ˆ r r/ G ∩ [0 , r ′ ] = ∅ the existence EIGHTED LENGTH 17 of a sequence { r k } in G ∩ [0 , r ′ ] such that lim k r k = 0 and 1 r k /r k +1
4. Nowif r k +1 r r k then L ([0 , r ] \ G ) r L ([0 , r k ] \ G ) r k +1 L ([0 , r k ] \ G ) r k ξ ( r k ) , according to (11). Therefore lim r → + L ([0 , r ] ∩ G ) r = 1 . Furthermore it follows from (10) thatlim sup r → + r ∈ G H ( C ∩ B ( x, r ))2 r . It is now clear that if G ∩ [0 , r ′ ] = ∅ then (9) holds. If instead G ∩ [0 , r ′ ] = ∅ thenEilenberg’s inequality implies that for each 0 < r r ′ one has3 r Z r card( C ∩ B ( x, ρ )) d L ( ρ ) H ( C ∩ B ( x, r )) . In particular Θ ∗ ( H C, x ) > / (cid:3) In the remaining part of this section we will obtain better information under theassumption that the gauge ξ is Dini.4.7. Theorem (Almost monotonicity) . —
Assume that: (A) C ⊆ X is compact and connected, U ⊆ X is open, r > ; (B) ξ is a Dini gauge with mean slope ζ ; (C) C is ( ξ, r ) almost minimizing in U ;It follows that for every x ∈ C ∩ U the function (0 , min { r , dist( x, bdry U ) } ) → R + : r exp[ ζ ( r )] H ( C ∩ B ( x, r ))2 r is nondecreasing.Proof. Fix x ∈ C ∩ U and let r ( x ) := dist( x, bdry U ). We define N ( ρ ) = card( C ∩ bdry B ( x, ρ )) ∈ N ∪ {∞} for 0 < ρ < r ( x ), and ϕ ( r ) = H ( C ∩ B ( x, r )) , for 0 < r < r ( x ). Notice that ϕ > A = C ∩ ( B ( x, b ) \ B ( x, a ))that Z ba N ( ρ ) d L ( ρ ) ϕ ( b ) − ϕ ( a )for every 0 < a < b < r ( x ). In particular N is almost everywhere finite and N ( ρ ) ϕ ′ ( ρ ) (12)at those ρ which are Lebesgue points of N and at which ϕ is differentiable. Since ϕ is nondecreasing this occurs almost everywhere. We next select 0 < ρ < r ( x ) suchthat N ( ρ ) < ∞ and we define C ′ = ( C \ B ( x, ρ )) ∪ [ y ∈ C ∩ bdry B ( x,ρ ) [[[ x, y ]]] . It follows from 4.3 that C ′ is a competitor for C in B ( x, ρ ). Assuming also that(12) holds, the almost minimizing property of C yields ϕ ( ρ ) = H ( C ∩ bdry B ( x, ρ )) (1 + ξ ( ρ )) H ( C ′ ∩ B ( x, ρ ))= (1 + ξ ( ρ )) N ( ρ ) ρ (1 + ξ ( ρ )) ϕ ′ ( ρ ) ρ . It follows from the above that ddρ log ϕ ( ρ ) = ϕ ′ ( ρ ) ϕ ( ρ ) > ξ ( ρ )) ρ > − ξ ( ρ ) ρ = ddρ log (cid:18) ρ exp[ − ζ ( ρ )] (cid:19) . Since this inequality occurs almost everywhere and ϕ is nondecreasing, we inferupon integrating each member thatlog ϕ ( r ) − log ϕ ( r ) > log (cid:18) r exp[ − ζ ( r )] (cid:19) − log (cid:18) r exp[ − ζ ( r )] (cid:19) . for every 0 < r < r < r ( x ). Our conclusion now easily follows. (cid:3) Corollary. —
At each x ∈ C ∩ U the density Θ ( H C, x ) = lim r → + H ( C ∩ B ( x, r ))2 r exists, and either Θ ( H C, x ) = 1 or Θ ( H C, x ) > / . Theorem (Lipschitz regularity) . —
Assume that: (A) C ⊆ X is compact and connected, U ⊆ X is open, < r < r , x ∈ C , B ( x, r ) ⊆ U , < τ / ; (B) ξ is a Dini gauge with mean slope ζ ; (C) C is ( ξ, r ) almost minimizing in U ; (D) exp[ ζ ( r )] τ / ; (E) ξ ( r ) τ / ; (F) card( C ∩ bdry B ( x, r )) = 2 .It follows that there exists τ r/ ρ τ r and a Lipschitz curve Γ such that C ∩ B ( x, ρ ) = Γ and C ∩ bdry B ( x, ρ ) consists of the two endpoints of Γ .Proof. We start arguing as in the proof of Theorem 4.6. Letting G and ϑ be definedas in that proof, we infer from our hypotheses (E) and (F) that ϑ ( r ) > − τ /
2, see(11), and we infer from our hypotheses (D), (E), (F), and the inequality τ / ζ ( r )] H ( C ∩ B ( x, r )) (1 + τ / r (1 + τ )2 r , (13)see (10). Thus there exists ρ ∈ G ∩ ( τ r/ , τ r ) which is a point of L approximatecontinuity of ρ card( C ∩ bdry B ( x, ρ )). It then follows from Theorem 4.4(F) thatour conclusion will be established provided we rule out the second alternative inthat conclusion. Assume if possible that such ˜ x ∈ C ∩ B ( x, ρ ) exists. The Eilenberginequality easily implies that Θ ( H C, ˜ x ) > /
2. The following contradiction
EIGHTED LENGTH 19 ensues:32 Θ ( H C, ˜ x ) exp[ ζ ((1 − τ ) r )] H ( C ∩ B (˜ x, (1 − τ ) r ))2(1 − τ ) r exp[ ζ ((1 − τ ) r )] H ( C ∩ B ( x, k x − ˜ x k + (1 − τ ) r ))2(1 − τ ) r exp[ ζ ((1 − τ ) r )] H ( C ∩ B ( x, k x − ˜ x k + (1 − τ ) r ))2( k x − ˜ x k + (1 − τ ) r ) (cid:18) k x − ˜ x k + (1 − τ ) r (1 − τ ) r (cid:19) which, according to k x − ˜ x k ρ τ r and Theorem 4.7, is bounded by exp[ ζ ( r )] H ( C ∩ B ( x, r ))2 r (cid:18) τ − τ (cid:19) which, according to (13), is bounded by (1 + τ ) (cid:18) τ − τ (cid:19) < τ / (cid:3) Definition. —
Let C ⊆ X and x ∈ C . We say that: (1) x is a regular point of C if for each δ > there exists < r < δ such that C ∩ B ( x, r ) is a Lipschitz curve Γ and C ∩ bdry B ( x, r ) consists of the twoendpoints of Γ ; (2) x is a singular point of C if it is not a regular point of C .The set of regular points of C is denoted reg( C ) , and the set of singular points of C is denoted sing( C ) = C \ reg( C ) . Theorem. —
Assume that: (A) C ⊆ X is compact and connected, U ⊆ X is open, r > ; (B) ξ is a Dini gauge; (C) C is ( ξ, r ) almost minimizing in U .It follows that U ∩ reg( C ) = U ∩ { x : Θ ( H C, x ) = 1 } , that U ∩ sing( C ) isrelatively closed in U ∩ C , and that H ( U ∩ sing( C )) = 0 .Proof. Let x ∈ U . We first show that if Θ ( H C, x ) = 1 then x is a regularpoint of C . Since C is closed, we infer that x ∈ C . If r ′ > r ′ H ( C ∩ B ( x, r ′ )) < r ′ ; the first inequality follows as in the proof ofTheorem 4.6(C), whereas the second results from our assumption. Therefore thereexists 0 < r < r ′ such that card( C ∩ bdry B ( x, r )) = 2, according to Eilenberg’sinequality. One can of course assume that r ′ is small enough for hypotheses (D)and (E) of Theorem 4.9 to be verified as well. It then follows from that Theoremthat C ∩ B ( x, r ′′ ) is indeed a Lipschitz curve, for some 0 < r ′′ < r ′ . Since r ′ isarbitrarily small, we conclude that x is a regular point of C .We now assume that x ∈ C is a regular point of C and we will establish thatΘ ( H C, x ) = 1. By definition, there are r > C ∩ bdry B ( x, r )) = 2. If we denote by x ,r and x ,r the corre-sponding two intersection points, then C ′ = ( C \ B ( x, r )) ∪ ([[[ x ,r , x ]]] ∪ [[[ x, x ,r ]]]) is a competitor for C in B ( x, r ), according to 4.3, and therefore H ( C ∩ B ( x, r )) (1 + ξ ( r ))2 r according to the almost minimizing property of C . If r is chosen smallenough for 1 + ξ ( r ) < (3 /
2) exp[ − ζ ( r )] then Θ ( H C, x ) < / ( H C, x ) = 1.We turn to proving the relative closedness of sing( C ) in U . We first observe thatthe function Θ : U → R : x Θ ( H C, x )is upper semicontinuous. Indeed, according to 4.7, Θ( x ) = inf r> Θ r ( x ), whereΘ r : U → R : x exp[ − ζ ( r )] H ( C ∩ B ( x, r ))2 r . It then suffices to note that U → R : x ( H C )( B ( x, r )) is uper semicon-tinuous, for each r >
0. Finally, U ∩ sing( C ) = U ∩ { x : Θ ( H C, x ) > / } ,according to 4.8, and the proof is complete.To conclude, since C is rectifiable [3, Theorem 4.4.8.] it follows from [10] thatΘ ( H C, x ) = 1 for H almost every x ∈ C . Hence H ( U ∩ sing( C )) = 0. (cid:3) The excess of length of a nonstraight path and a regularitytheorem . —
In this section X denotes a uniformly rotund Banach space, with modulus of uniform rotundity δ X ,recall 2.10.If x , x and z are the three vertices of a nondegenerate triangle in a Hilbert space ,then the length of the broken line from x to x passing through z is substantiallylarger than the length of the straight path from x to x , specifically k x − z k + k z − x k > k x − x k s h max {k x − z k , k z − x k } , where h = dist( z, L ), L = x + span { x − x } . This ensues from the PythagoreanTheorem and from the observation that among all such triangles with same height h , the isoceles triangle has the shortest perimeter.If X is an arbitrary Banach space, the collection of inequalities k x − z k + k z − x k > k x − x k (cid:18) δ (cid:18) h max {k x − z k , k z − x k} (cid:19)(cid:19) , for some gauge δ , is equivalent to the uniform convexity of X , see [7, LemmaIV.1.5]. Next comes an ersatz of the Pythagorean Theorem that makes this obser-vation quantitative, showing that δ and the modulus of uniform convexity of X have the same asymptotic behavior.5.2. Proposition. —
Assume that X is a uniformly convex Banach space withmodulus of uniform convexity δ X , and that x , x , z are the vertices of a nondegen-erate triangle. It follows that k x − z k + k z − x k > k x − x k (cid:18) δ X (cid:18) dist( z, x + span { x − x } )2( k x − z k + k z − x k ) (cid:19)(cid:19) . Proof.
We let S X = bdry B (0 ,
1) denote the unit sphere. Referring to the Hahn-Banach Theorem, with each unit vector v ∈ S X we associate a closed linear subspace H v ⊆ X of codimension 1 such that B X lies entirely on one side of v + H v . Weobserve that v H v by necessity, and that there is no restriction to assume that H − v = H v .We denote by V the 2 dimensional affine subspace of X containing x , x and z , and V the corresponding linear subspace. We define v = k x − x k − ( x − x ), EIGHTED LENGTH 21 L v = V ∩ H v , and L ′ v = span { v } . As L v and L ′ v are nonparallel, the followingdefines y ′ : { y ′ } = ( x + L ′ v ) ∩ ( z + L v ) . We now distinguish between the cases when y ′ lies on x + L ′ v between x and x ,or not. First case : y ′ ∈ [[[ x , x ]]] . We define ρ = k x − y ′ k , B = V ∩ B ( x , ρ ), andwe let S be the boundary of B relative to V . Our choice of H v guarantess that B lies, in V , on one side of y ′ + L v . Therefore, among the two points of which S ∩ ( x + span { z − x } ) consists, one belongs to [[[ x , z ]]]. We denote it as y . b bb b b x y ′ x S B zy z + L v x + L ′ v v Figure 2.
Situation of the first case.We now abbreviate ε = ρ − k y ′ − y k and we infer from the uniform convexity of X that (cid:13)(cid:13)(cid:13)(cid:13) y ′ + y (cid:13)(cid:13)(cid:13)(cid:13) = (1 − β ) ρ (1 − δ X ( ε )) ρ , (14)where the equality defines β (thus β > δ X ( ε )).We claim that k y − z k > βρ . (15)In other words, we are comparing the lengths of the line segments [[[ y, z ]]] and [[[( y ′ + y ) / , w ]]], where w is at the intersection of x + span { ( y ′ + y ) / − x } and S , in thetriangle of vertices y ′ , y and z (see Figure 3 just after).With each s ∈ R we associate z s = y ′ + s ( z − y ′ ). If we denote by P : V → V theprojection onto the line y ′ + span { y ′ − x } , parallel to the line y ′ + span { z − y ′ } ,then the maps f s = (cid:0) P ↾ [[[ x ,z s ]]] (cid:1) − are affine bijections from [[[ x , y ′ ]]] to [[[ x , z s ]]]. Wenote that the convex function s
7→ k z s − x k has a mimimum at s = 0 – accordingto our choice of H v –, and therefore is nondecreasing on the interval s >
0. Itfollows that there exists a nondecreasing function s λ ( s ), s >
0, such that H ( f s ( I )) = λ ( s ) H ( I )for each interval I ⊆ [[[ x , y ′ ]]]. We now choose 0 s s such that ( y ′ + y ) / ∈ [[[ x , z s ]]] and z = z s . It is easily seen that P ([[[ y, z ]]]) ⊇ P ([[[( y ′ + y ) / , z s ]]]) ⊇ P ([[[( y ′ + y ) / , w ]]])and the the proof of (15) follows. b x z y ′ y b bb ρ βρ S Figure 3.
Illustration of (15): the blue segment is larger than thetiny red one.It ensues from (15) that k x − z k = k x − y k + k y − z k > (1 + β ) k x − y k > (1 + δ X ( ε )) k x − y k . (16)We are now going to establish that k x − z k > (cid:18) δ X (cid:18) dist( z, x + span { x − x } )2 k x − z k (cid:19)(cid:19) k x − y ′ k . (17)Either k x − z k > k x − y k , in which case (17) readily holds since δ X ( η ) < η
2, or k x − z k < k x − y k . In the latter case,dist( z, x + span { x − x } ) = k z − x kk y − x k dist( y, x + span { x − x } ) y, x + span { x − x } ) k y − y ′ k , therefore ε = k y − y ′ k ρ > dist( z, x + span { x − x } )2 k x − y k > dist( z, x + span { x − x } )2 k x − z k . Thus (17) follows from (16), because δ X is nondecreasing.We now repeat the same argument with the vertex x playing the role of x .Since L − v = L v , we observe that the new point y ′ coincides with the one foundpreviously. The analogous calculations therefore yield k x − z k > (cid:18) δ X (cid:18) dist( z, x + span { x − x } )2 k x − z k (cid:19)(cid:19) k x − y ′ k . (18)Upon summing inequalities (17) and (18) we obtain k x − z k + k z − x k > (cid:18) δ X (cid:18) dist( z, x + span { x − x } )2 max {k x − z k , k z − x k} (cid:19)(cid:19) k x − x k , (19) EIGHTED LENGTH 23 which proves the proposition in this case.
Second case : y ′ [[[ x , x ]]] . We start by observing that we may as well assume λ := min {k x − z k , k z − x k} < k x − x k (20)for otherwise the conclusion k x − z k + k z − x k > k x − x k > (cid:18) δ X (cid:18) dist( z, x + span { x − x } )2( k x − z k + k z − x k ) (cid:19)(cid:19) k x − x k readily follows from the trivial inequality 0 < δ X ( η )
1, 0 < η
2. We define c = ( x + x ) / r = k x − x k /
2, and we let { x ′ , x ′ } = ( x + L ′ v ) ∩ bdry B ( c, r + λ ) . It follows from our choice of λ , (20), and the definition of L v that the point y ′ defined by { y ′ } = ( x + L ′ v ) ∩ ( z + L v )belongs to the line segment [[[ x ′ , x ′ ]]]. Therefore the first case of this proof applies tothe triangle with vertices x ′ , x ′ and z . Accordingly, k x ′ − z k + k z − x ′ k > (1 + δ ′ ) k x ′ − x ′ k (21)where δ ′ = δ X (cid:18) dist( z, x + span { x − x } )2 max {k x ′ − z k , k z − x ′ k} (cid:19) . We notice that k x ′ − z k + k z − x ′ k k x − z k + k z − x k + 2 λ k x − z k + k z − x k + 2(1 + δ ′ ) λ as well as k x ′ − x ′ k = k x − x k + 2 λ . Plugging these inequalities in (21) yields k x − z k + k z − x k > (1 + δ ′ ) k x − x k , and it remains to observe that δ ′ > δ X (cid:18) dist( z, x + span { x − x } )2( k x − z k + k z − x k ) (cid:19) becausemax {k x ′ − z k , k z − x ′ k} max {k x − z k , k z − x k} + λ = k x − z k + k z − x k . The proof is now complete. (cid:3)
The following is an ersatz of the Pythagorean Theorem, valid in uniformly convexBanach spaces.5.3.
Proposition (Height bound) . —
Assume that: (A) C ⊆ X is compact, connected, U ⊆ X is open, < r < r , x ∈ C , B ( x, r ) ⊆ U ; (B) ξ is a gauge and ξ ( r ) δ X (1 / ; (C) C is ( ξ, r ) almost minimizing in U ; (D) C ∩ B ( x, r ) is a Lipschitz curve with endpoints x and x , and C ∩ bdry B ( x, r ) = { x , x } ; (E) L = span { x − x } .It follows that (F) k x − x k > (1 − ξ ( r ))2 r ; (G) For every z ∈ C ∩ B ( x, r ) one has dist( z, x + L ) δ − X ◦ ξ )( r ) . Remark. —
Under the same assumptions one can in fact show that1 r dist H (cid:2) C ∩ B ( x, r ) , ( x + L ) ∩ B ( x, r ) (cid:3) δ − X ◦ ξ )( r ) , but only the weaker version (G) will be used in the proof of Theorem 5.5. Proof of Proposition 5.3.
We start by observing that2 r H ( C ∩ B ( x, r )) (1 + ξ ( r )) k x − x k . (22)The first inequality results from 4.4(D) as in the proof of 4.6(C), and the secondinequality follows from the almost minimizing property of C together with the factthat C ′ = ( C \ B ( x, r )) ∪ [[[ x , x ]]]is a competitor for C in B ( x, r ), according to 4.3. This proves conclusion (F).Let z ∈ C ∩ B ( x, r ) and define h z = dist( z, x + L ) . Notice that k x − z k + k z − x k r , name Γ the Lipschitz curve C ∩ B ( x, r ), andwrite Γ (resp. Γ ) for the subcurve of Γ with endpoints x and z (resp. z and x ).It ensues from 5.2 that k x − x k (cid:18) δ X (cid:18) h z r (cid:19)(cid:19) k x − z k + k z − x k H (Γ ) + H (Γ )= H ( C ∩ B ( x, r )) (1 + ξ ( r )) k x − x k where the last inequality follows from (22). Therefore δ X (cid:18) h z r (cid:19) ξ ( r ) , and in turn, h z r δ − X ◦ ξ )( r ) , (23)recall 2.10.We abbreviate η = 8 r ( δ − X ◦ ξ )( r ). So far we showed that given z ∈ C ∩ B ( x, r ),there is v z ∈ L such that k z − ( v z + x ) k η . As x ∈ C ∩ B ( x, r ), there exists v x ∈ L such that k x − ( v x + x ) k η . Therefore k z − ( v z − v x + x ) k k z − ( v z + x ) k + k x − ( v x + x ) k η and the proof of (G) is complete. (cid:3) In the following we use the terminology universal constant for real numbers thatdo not depend on the data ( X , C , ξ etc).5.5. Theorem ( C regularity) . — There are universal constants η > and C > with the following property. Assume that: (A) C ⊆ X is compact, connected, U ⊆ X is open, r > , x ∈ C , B ( x , r ) ⊆ U ; (B) ξ is a gauge and the gauge δ − X ◦ ξ is Dini ; (C) C is ( ξ, r ) almost minimizing in U ; (D) exp[ ζ ( r )] η where ζ is the mean slope of ξ ; (E) ( δ − X ◦ ξ )( r ) η ; (F) H ( C ∩ B ( x , r )) (1 + η )2 r . EIGHTED LENGTH 25
It follows that C ∩ B ( x , η r ) is a C curve Γ . Furthermore if γ is an arclengthparametrization of Γ then osc( γ ′ ; η ) C ω ( C η ) where ω is the mean slope of the Dini gauge δ − X ◦ ξ . Remark. —
Several comments are in order.(A) Since δ X ( ε ) ε for every 0 < ε <
2, see [11, p. 63], we infer that t √ t δ − X ( t ) whenever 0 < t
1. Thus for any gauge ξ and any t suchthat ξ ( t ) ξ ( t ) δ − X ( ξ ( t )) , and it immediately follows from the definition that ξ is a Dini gauge when-ever δ − X ◦ ξ is Dini. In particular hypothesis (D) makes sense. In ourstatement ζ is the mean slope of ξ and ω is the mean slope of δ − X ◦ ξ .(B) Under the assumptions of the Theorem, if 0 < τ < x ∈ C ∩ B ( x , τ r )and 0 < r (1 − τ ) r then 4.7 (almost monotonicity) implies that H ( C ∩ B ( x, r ))2 r exp[ ζ ((1 − τ ) r ] H ( C ∩ B ( x, (1 − τ ) r ))2(1 − τ ) r exp[ ζ ( r )] H ( C ∩ B ( x , r ))2 r (cid:18) − τ (cid:19) (1 + η ) − τ . In particular, when τ is small, a version of hypothesis (F) holds (with aslightly worse constant that η ) for x close to x and r sligthly smaller than r . We will refer to this observation in the core of the proof.(C) It is useful to notice that the Theorem applies at all. In fact, if C ⊆ X iscompact, connected and H ( C ) < ∞ , and if ξ is a Dini gauge, then for H almost every x ∈ C there exists r = r ( x ) > r small enough independently of x , whereas assumption (F)follows from the rectifiability of C [3, Theorem 4.4.8.] which implies thatΘ ( H C, x ) = 1 for H almost every x ∈ C , see e.g. [10]. Thus theonly nontrivial assumption of the Theorem, apart from the almost mini-mizing property of C , is that δ − X ◦ ξ be Dini.(D) Let us spell out the kind of regularity obtained in case ξ ( r ) Cr α , 0 < α
1, and X = L p , 1 < p < ∞ . Here we consider an L p space relative to anymeasure space, and we point out that the ( ξ, r ) almost minimizing propertyis verified by a solution of the variational problem ( P F,w ) (see near the endof section 3) provided the weight w is H¨older continuous of exponent α (see3.8). One infers from [11, p.63] that δ − L p ( ε ) C p ε / max { ,p } . Thus thegauge δ − L p ◦ ξ is geometric and it follows that ˚Γ is C ,α/ max { ,p } . Proof of Theorem 5.5.
We say a pair ( x, r ) ∈ C × ( R + \ { } ) is good if C ∩ B ( x, r )is a Lipschitz curve Γ x,r and if C ∩ bdry B ( x, r ) consists exactly of the endpointsof Γ x,r . Claim
For every x ∈ C ∩ B ( x , r / and every < r (5 / r , thereexists r/ ρ r/ such that ( x, ρ ) is good. We first notice that H ( C ∩ B ( x, (5 / r )) H ( C ∩ B ( x , r )) (1 + 1 / r , according to hypothesis (F). Thus we infer from hypothesis (D) and 4.7 (almostmonotonicity) that H ( C ∩ B ( x, r ))2 r exp[ ζ ( r )] H ( C ∩ B ( x, r ))2 r exp[ ζ ((5 / r )] H ( C ∩ B ( x, (5 / r ))2(5 / r < (cid:18) (cid:19) (cid:18) (cid:19) = 32 − . We next define an L measurable set G = [0 , r ] ∩ { ρ : card C ∩ bdry B ( x, ρ ) = 2 } ,recall 2.7. It follows from 4.4(D) and Eilenberg’s inequality that2 L ( G ) + 3( r − L ( G )) Z r (card C ∩ bdry B ( x, ρ )) d L ( ρ ) H ( C ∩ B ( x, r )) < (cid:18) − (cid:19) r . It readily follows that L ( G ) > r/
12. Pick r ′ ∈ G with r ′ > r/
12 and apply 4.9(with τ = 1 / (cid:4) We apply
Claim to x = x and r = (5 / r to find some ρ such that1144 (cid:18) r (cid:19) ρ (cid:18) r (cid:19) and C ∩ B ( x , ρ ) is a Lipschitz curve Γ .We define r j = 72 − j , j ∈ N , and for the remaining part of this proof we assume j > j where j is chosen sufficiently large for r j (5 / r . For such r j and x ∈ C ∩ B ( x , ρ ), Claim applies to yield ρ x,j such that r j +1 r j ρ x,j r j r j +1 (24)and C ∩ B ( x, ρ x,j ) is a Lipschitz curve Γ x,j whose two endpoints coincide with C ∩ bdry B ( x, ρ x,j ). We easily infer that3 ρ x,j ρ x,j +1 . (25)We parametrize Γ by arclength γ : [ a, b ] → X . Corresponding to each good pair( x, ρ x,j ) obtained above, we notice Γ x,j ⊆ Γ , and thereforeΓ x,j = γ ([ s x + h − x,j , s x + h + x,j ])where a < s x < b is so that x = γ ( s x ) and h − x,j < < h + x,j . Claim
One has ρ x,j | h ± x,j | (1 + 2 ξ ( r j )) ρ x,j (26) and / | h ± x,j || h ± x,j +1 | . (27)The first of the four (set of) inequalities simply follows from Lip γ ρ x,j = k γ ( s x + h ± x,j ) − γ ( s x ) k | h ± x,j | . We next infer from the almost minimizing property of C and the fact that ( x, ρ x,j )is a good pair, | h − x,j | + | h + x,j | = H (Γ − x,j ) + H (Γ + x,j ) = H (Γ x,j )= H ( C ∩ B ( x, ρ x,j )) (1 + ξ ( ρ x,j ))2 ρ x,j . The second inequality ensues easily from this and from the first inequality.
EIGHTED LENGTH 27
The third and fourth inequalities are consequences of (25) and of | h ± x,j | ρ x,j which itself follows from the second inequality and ξ ( r j ) / (cid:4) Next we apply 5.3 to each good pair ( x, ρ x,j ), x ∈ C ∩ B ( x , ρ ). This providesus with a 1 dimensional linear subspace L x,j ⊆ X such thatmax { dist( z, x + L x,j ) : z ∈ Γ x,j } ε ( r j ) ρ x,j , where we have abbreviated ε ( r ) = 16( δ − X ◦ ξ )( r ) . It is useful to recall that ξ ε , cf. 5.6. Associated with each h ∈ [ h − x,j , h − x,j +1 ] ∪ [ h + x,j +1 , h + x,j ] we choose v x,h,j ∈ L x,j such that k γ ( s x + h ) − x − v x,h,j k ε ( r j ) ρ x,j . (28)We choose a unit vector w x,j ∈ X spanning L x,j , and t x,h,j ∈ R such that v x,h,j = t x,h,j w x,j . Replacing w x,j by − w x,j if necessary, we also assume that t x,h + x,j ,j > v ± x,j = v x,h ± x,j ,j t ± x,j = t x,h ± x,j ,j . B ( x, ρ x,j ) b L x,j w x,j C b bb v + x,j = t + x,j w x,j γ ( s x + h + x,j ) ε ( r j ) ρ x,j x = γ ( s x ) b v − x,j = t − x,j w x,j γ ( s x + h − x,j )Γ x,j Figure 4.
Notation of the proof of differentiability.
Claim
The following hold: (cid:12)(cid:12) | t ± x,j | − ρ x,j (cid:12)(cid:12) ε ( r j ) ρ x,j , (29) and t + x,j > as well as t − x,j < . Since k γ ( s x + h ± x,j ) − x k = ρ x,j the first inequality is an immediate consequence of(28). In order to determine the signs of the t ± x,j we proceed as follows. As t + x,j > w x,j ) we infer from the first conclusion of the claim that t + x,j > t − x,j <
0. We infer from (28) that k γ ( s x + h + x,j ) − γ ( s x + h − x,j ) − ( t + x,j − t − x,j ) w x,j k ε ( r j ) ρ x,j . (30) It follows from 5.3(F) that k γ ( s x + h + x,j ) − γ ( s x + h − x,j ) k > ρ x,j (1 − ξ ( r j )) . (31)Thus, | t + x,j − t − x,j | > ρ x,j (1 − ε ( r j )) , (32)according to (30) and (31). If t + x,j and t − x,j had the same sign it would follow fromthe first conclusion of this claim that | t + x,j − t − x,j | ε ( r j ) ρ x,j . Plugging this into (32) would yield2(1 − ε ( r j )) ε ( r j ) , in contradiction with hypothesis (E). (cid:4) We now introduce a notation for the difference quotients of γ . Let s ∈ [ a, b ] and h ∈ R \ { } such that s + h ∈ [ a, b ]. We define △ γ ( s, h ) = γ ( s + h ) − γ ( s ) h . We will prove that △ γ ( s x , h ) and △ γ ( s x , h + x,j ) are close, for h ∈ [ h − x,j , h − x,j +1 ] ∪ [ h + x,j +1 , h + x,j ]. To this end we will observe these two vectors are close to positivemultiples of w x,j , and they both have length nearly equal to 1. Claim
For every x ∈ C ∩ B ( x , η r ) , every j > j , and every h ∈ [ h − x,j , h − x,j +1 ] ∪ [ h + x,j +1 , h + x,j ] one has − C ξ ( r j ) k△ γ ( s x , h ) k , for some universal constant C > . The second inequality simply follows from the fact that Lip γ
1. In order toestablish the first inequality we abbreviate x − j = γ ( s x + h − x,j ), y = γ ( s x + h ), and x + j = γ ( s x + h + x,j ). We will also denote by Γ p,q the portion of the curve Γ x,j withendpoints p, q ∈ Γ x,j . We first show that there exists a universal η > η | h | k γ ( s x + h ) − γ ( s x ) k = k y − x k (33)for every h ∈ [ h − x,j , h − x,j +1 ] ∪ [ h + x,j +1 , h + x,j ]. We note that for such h , H (Γ x,y ) = | h | > ρ x,j +1 > ρ x,j , where we used the fact that γ is parametrized by arclength, inequalities (26) (ap-plied to j + 1) and (25). Now assume if possible that k x − y k < η | h | for some small η to be determined momentarily. We would then infer that H (Γ y,x + j ) > k y − x + j k > ρ x,j − k x − y k > (1 − η ) ρ x,j , according to 3.2 and (26). In turn, H (Γ x,x + j ) = H (Γ x,y )+ H (Γ y,x + j ) > ρ x,j (cid:18) − η (cid:19) > ρ x,j (cid:18) (cid:19) if η is small enough. Of course the same estimate holds with x − j in place of x + j .Therefore, 2 ρ x,j (cid:18) (cid:19) H (Γ x − j ,x ) + H (Γ x,x + j )= H ( C ∩ B ( x, ρ x,j )) (cid:18) (1 + η ) − η (cid:19) ρ x,j , (34) EIGHTED LENGTH 29 according to Remark 5.6(B) applied with τ = η and r = ρ x,j . One can choose η small enough to yield a contradiction, thereby proving the validity of (33).We next improve on (33) by showing that there exists a universal C > | h | C ξ ( r j ) k γ ( s x + h ) − γ ( s x ) k = k y − x k . (35)We define 0 η ( h ) by the the following first equation k y − x k = | h | η ( h ) = H (Γ x,y )1 + η ( h )(the second equation ensues from the fact that γ is an arclength parametrization)and we seek an upper bound for η ( h ). We define set S ′ ⊆ B ( x, ρ x,j ) by S ′ = Γ x − j ,x ∪ [[[ x, y ]]] ∪ Γ y,x + j . We notice that S ′ is the image of a (possibly non injective) Lipschitz map [ − , → B ( x, ρ x,j ) that sends − x − j and +1 to x + j . It is now easy to check that C ′ = ( C \ B ( x, ρ x,j )) ∪ S ′ is a competitor for C in B ( x, ρ x,j ). Therefore H ( C ∩ B ( x, ρ x,j )) (1 + ξ ( r j )) H ( C ′ ∩ B ( x, ρ x,j )) . (36)Furthermore, H ( C ′ ∩ B ( x, ρ x,j )) H (Γ x − j ,x ) + k x − y k + H (Γ x,x + j ) . (37)Since also H ( C ∩ B ( x, ρ x,j )) = H (Γ x,j ) = H (Γ x − j ,x ) + H (Γ x,y ) + H (Γ x,x + j ) , it follows from the definition of η ( h ) that H ( C ∩ B ( x, ρ x,j )) − H ( C ′ ∩ B ( x, ρ x,j )) = H (Γ x,y ) − k x − y k > η ( h ) k y − x k > η ( h ) η | h | > η ( h ) η | h ± x,j +1 | > η ( h ) η ρ x,j +1 according to (33) and (26) (applied to j + 1). Plugging this into (36) we obtain η ( h ) η ρ x,j +1 ξ ( r j ) H ( C ′ ∩ B ( x, ρ x,j ) . Finally, we notice that (37) and (34) imply that H ( C ′ ∩ B ( x, ρ x,j )) H ( C ∩ B ( x, ρ x,j )) ρ x,j (if η is small enough). Thus, η ( h ) ρ x,j ρ x,j +1 η ξ ( r j ) , which proves (35) in view of (25). (cid:4) Claim
Let x ∈ C ∩ B ( x , η r ) and j > j . If h − x,j h h − x,j +1 then t x,h,j < and if h + x,j +1 h h + x,j then t x,h,j > . We prove it in case h + x,j +1 h h + x,j , the other case is analogous. We inferfrom (28) that k γ ( s x + h + x,j ) − γ ( s x + h ) − ( t + x,j − t x,h,j ) w x,j k ε ( r j ) ρ x,j , and in turn from (26) (applied to both j and j + 1) that | t + x,j − t x,h,j | k γ ( s x + h + x,j ) − γ ( s x + h ) k + 2 ε ( r j ) ρ x,j h + x,j − h + 2 ε ( r j ) ρ x,j h + x,j − h + x,j +1 + 2 ε ( r j ) ρ x,j (1 + 2 ε ( r j )) ρ x,j − ρ x,j +1 + 2 ε ( r j ) ρ x,j . Assuming if possible that t x,h ρ x,j (1 − ε ( r j )) + | t x,h,j | t + x,j + | t x,h,j | = | t + x,j − t x,h,j | . Thus, | t x,h,j | ε ( r j ) ρ x,j − ρ x,j +1 (5 . ε ( r j ) − ρ x,j +1 (5 . . η − ρ x,j +1 . Since the right member of the inequality is negative if η is chosen small enough, weobtain the sought for contradiction. (cid:4) We are now ready to finish off the proof of the theorem. We start by fixing x ∈ B ( x , η r ) and we will show that γ is differentiable at s x . To this end we fixfirst j > j and h ∈ [ h − x,j , h − x,j +1 ] ∪ [ h + x,j +1 , h + x,j ]. Dividing (28) by | h | , and referringto (26) and (25), we obtain (cid:12)(cid:12)(cid:12)(cid:12) k△ γ ( s x , h ) k − (cid:12)(cid:12)(cid:12)(cid:12) t x,h,j h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13) △ γ ( s x , h ) − t x,h,j h w x,j (cid:13)(cid:13)(cid:13)(cid:13) ε ( r j ) ρ x,j | h | ε ( r j ) ρ x,j | h ± x,j +1 | ε ( r j ) ρ x,j ρ x,j +1 ε ( r j ) . (38)We notice that t x,h,j /h and t + x,j /h + x,j are both positive according to Claims and
Claim that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t x,h,j h − t + x,j h + x,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t x,h,j h (cid:12)(cid:12)(cid:12)(cid:12) − k△ γ ( s x , h ) k (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) k△ γ ( s x , h ) k − k△ γ ( s x , h + x,j ) k (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k△ γ ( s x , h + x,j ) k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t + x,j h + x,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ε ( r j ) + C ξ ( r j )= C ′ ε ( r j )for some universal constant C ′ >
0. In view of (38) we thus obtain k△ γ ( s x , h + x,j ) − △ γ ( s x , h ) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) △ γ ( s x , h + x,j ) − t + x,j h + x,j w x,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t + x,j h + x,j − t x,h,j h ! w x,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) t x,h,j h w x,j − △ γ ( s x , h ) (cid:13)(cid:13)(cid:13)(cid:13) . ε ( r j ) + C ′ ε ( r j )= C ′′ ε ( r j ) . (39)Applying temporarily this inequality to h = h + x,j +1 we see that k△ γ ( s x , h + x,j +1 ) − △ γ ( s x , h + x,j ) k C ′′ ε ( r j ) . Thus for any k > j and any l > k△ γ ( s x , h + x,k + l ) − △ γ ( s x , h + x,k ) k k + l − X j = k k△ γ ( s x , h + x,j +1 ) − △ γ ( s x , x + x,j ) k (16 . / C ′′ ω ( r k − ) =: C ′′′ ω ( r k − ) . (40)according to 2.9. Therefore {△ γ ( s x , h + x,j ) } j is a Cauchy sequence, whence alsoconvergent, and we denote its limit by γ ′ ( s x ). In order to verify that γ ′ ( s x ) is EIGHTED LENGTH 31 the derivative of γ at s x we combine (39) with (40) in which we let l → ∞ : If h ∈ [ h − x,j , h − x,j +1 ] ∪ [ h + x,j +1 , h + x,j ] then k γ ′ ( s x ) − △ γ ( s x , h ) k k γ ′ ( s x ) − △ γ ( s x , h + x,j ) k + k△ γ ( s x , h + x,j ) − △ γ ( s x , h ) k C ′′ ε ( r j ) + C ′′′ ω ( r j − ) C ′′′ ω ( r j − )= C iv ω ( r j − ) . (41)We notice that (24), (25) and (26) imply that r j − . | h | . Thus (41) finallyyields k γ ′ ( s x ) − △ γ ( s x , h ) k C v ω ( C v | h | ) , thereby establishing the differentiability of γ at h .Finally, if x , x ∈ C ∩ B ( x , η r ) we let s = s x and s = s x and h = H (Γ x ,x ) = | s − s | (since γ is parametrized by arclength). Upon noticing that △ γ ( s , h ) = △ γ ( s , − h ) we infer from the above inequality that k γ ′ ( s ) − γ ′ ( s ) k k γ ′ ( s ) − △ γ ( s , h ) k + k△ γ ( s , − h ) − γ ′ ( s ) k C v ω ( C v | h | ) C ω ( C | s − s | )for some universal constant C >
0, and the proof of the theorem is complete. (cid:3) Application to the quasihyperbolic distance . —
In 6.2 X is an arbi-trary Banach space, in 6.3 X is a reflexive Banach space, and in 6.4 X is uniformlyrotund and δ − X is Dini.6.2 (The quasihyperbolic distance) . — In this section we assume D ⊆ X is anonempty open subset with the following property: For any pair x, y ∈ D thereexists a curve Γ ⊆ D with endpoints x and y , and H (Γ) < ∞ . We now proceedto define on D a new metric d q , so-called the quasihyperbolic distance of D . Westart by abbreviating h : D → R : x dist( x, X \ D ) . Since D is open and bounded we notice that 0 < h ( x ) < ∞ , x ∈ D . It is alsohelpful to note, for further purposes, that 1 /h is locally Lipschitzian. In fact, if η > D η := D ∩ { x : h ( x ) > η } then Lip(1 /h ) ↾ D η η − , as follows from (cid:12)(cid:12)(cid:12)(cid:12) h ( x ) − h ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) h ( y ) − h ( x ) h ( y ) h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) | h ( y ) − h ( x ) | η k y − x k η ,x, y ∈ D η . For x , y ∈ D we now define d q ( x , y ) = inf (cid:26) Z Γ h d H : Γ ⊆ D is a curve of finite lengthwith endpoints x and y (cid:27) . It is obvious that for each given competitor Γ one has 1 /h ∈ L ( X, H Γ); inparticular d q ( x , y ) < ∞ . There are two points to this section:(A) The infimum in the definition of d q ( x , y ) is achieved by some curve Γprovided X is a separable reflexive Banach space and D is convex;(B) If Γ is a curve that achieves the infimum in the definition of d q ( x , y ) thenΓ \ { x , y } is a C open curve provided X is uniformly rotund and δ − X isDini. In case X = ℓ n is a finite dimensional Euclidean space, both results have beenobtained by G. Martin [12]. J. V¨ais¨al¨a proves the existence part (A), [16], andobtains the regularity part in case X = ℓ is Hilbert, [17]. It should be noted thatour method shows a quasihyperbolic geodesic in ℓ is C , / in the interior, when infact the stronger C , regularity entails from its quasihyperbolic property, see [17].6.3 (Existence of quasihyperbolic geodesics) . — We assume X is reflexive. Asmentioned before, the existence part (A) above was proved by J. V¨ais¨al¨a. We nowbriefly comment on the proof. In order that 3.4 apply to a minimizing sequence { Γ n } , we ought to show that(1) the weight w : X → (0 , + ∞ ] defined by w ( x ) = 1 /h ( x ) if x ∈ D and w ( x ) = + ∞ otherwise, is weakly* lower semicontinuous, i.e. weakly lowersemicontinuous since X is reflexive;(2) there exists R > n ⊆ B ( x , R ) for every n .The Hahn-Banach Theorem implies that clos D = ∩ E where E is the collec-tion of translates z + E of closed half spaces E ⊆ X such that D ⊆ z + E .Therefore h ( x ) = inf { dist( x, E ) : E ∈ E } , and establishing (1) boils down toshowing that each ψ E : x dist( x, E ) is weakly upper semicontinuous. In fact,as H = bdry E is a hyperplane, there exists a linear form x ∗ : X → R suchthat dist( x, bdry E ) = |h x, x ∗ i| . Since this function of x is Lipschitzian, one has x ∗ ∈ X ∗ , and the weak continuity of ψ E now readily follows. To establish (2) weassume that M := sup n R Γ n w d H < ∞ , and we write L n := H (Γ n ). Consid-ering an arclength parametrization γ n : [0 , L n ] → X of Γ n with γ n (0) = x , weestimate Z Γ n w d H = Z L n h ( γ n ( t )) dt > Z L n h ( x ) + k γ n ( t ) − γ n (0) k dt > Z L n h ( x ) + t dt = log (cid:18) L n h ( x ) (cid:19) . Whence L n h ( x ) e M . Since k x − x k L n for every x ∈ Γ n , item (2) is proved.See [16] for details.We now apply 3.3 and 3.4 to obtain a connected compact set C ⊆ clos D and a(not relabeled) subsequence still denoted { Γ n } , such that dist ∗ H (Γ n , C ) → R C w d H d q ( x , y ). Recall 2.8 that C contains a curve Γ with endpoints x and y ; thus R Γ w d H d q ( x , y ). It remains to establish that Γ ∩ bdry D = ∅ .For each 0 < η < min { h ( x ) , h ( y ) } we note that if Γ ∩ ( D \ D η ) = ∅ then H (Γ ∩ ( D η \ D η )) > η and therefore R Γ ∩ ( D η \ D η ) w d H > . As R Γ w d H < ∞ ,Γ can therefore meet at most finitely many annuli D − j \ D − j +1 .6.4 (Regularity of quasihyperbolic geodesics) . — We assume X is uniformly rotundand δ − X is Dini. Letting Γ be a quasihyperbolic geodesic with endpoints x and y ,we recall from the argument above that Γ ⊆ U = D η for some η >
0, and also thatLip w ↾ D η η − . It follows from 3.8 that 5.5 will apply with ξ ( r ) Cr at points x ∈ Γ such that Θ ( H Γ , x ) = 1. This is the case of each x ∈ ˚Γ, according to3.9. Thus Γ is, near x , a C ,ω curve where ω is the mean slope of δ − X ◦ ξ . Since ξ ( r ) Cr , ω is asymptotic to the mean slope of δ − X . See also 5.6(D).7. Differentiability of almost minimizing curves in 2 dimensionalrotund spaces . —
In this section X is 2dimensional and (uniformly) rotund. We also assume that its norm x
7→ k x k is C smooth on X \ { } . We let S X = X ∩ { v : k v k = 1 } be the unit circle. EIGHTED LENGTH 33 . —
Here we localize (in direction) the definitionof modulus of uniform rotundity of X . Specifically, for v ∈ S X and 0 < ε
1, wedefine δ X ( v ; ε ) = inf (cid:26) − (cid:13)(cid:13)(cid:13)(cid:13) v + ( v + h )2 (cid:13)(cid:13)(cid:13)(cid:13) : h ∈ X, k v + h k k h k > ε (cid:27) . Our first aim is to show that the rotundity and C smoothness of the norm imply δ X ( v ; ε ) has the best possible behavior (i.e. is asymptotic to ε ) except possibly fora closed nowhere dense set of directions v .7.3. Theorem. —
There exists G ⊆ S X with the following properties. (A) S X \ G is closed and has empty interior in S X ; (B) For every v ∈ G there exists ρ > and C > such that for each v ∈ S X ,if k v − v k < ρ then δ − X ( v ; ε ) C √ ε for every < ε .Proof. We denote the norm of X as f : X → R : v
7→ k v k . Given v ∈ S X we choosea unit vector e v ∈ X that generates the tangent line T v S X . Thus v, e v is a basis of X and we denote as v ∗ , e ∗ v the corresponding dual basis (i.e. x = h x, v ∗ i v + h x, e ∗ v i e v for each x ∈ X ). We observe that Df ( v ) = v ∗ and that D f ( v ) is a positivesemidefinite bilinear form on X (owing to the convexity of f ), k v + h k = 1 + h h, v ∗ i + 12 D f ( v )( h, h ) + o ( k h k ) (42)whenever h ∈ X , where o ( k h k ) is little o of k h k uniformly in v .Upon noticing that the function S X → R : v D f ( v )( e v , e v )is continuous, we immediately infer that G = S X ∩ { v : D f ( v )( e v , e v ) > } is an open subset of S X and that B := S X \ G = S X ∩ { v : D f ( v )( e v , e v ) = 0 } . Assume if possible that B contains a nonempty open interval V ⊆ S X . For each v ∈ V one has D f ( v )( e v , e v ) = 0 and D f ( v )( v, v ) = 0 (because the norm f ishomogeneous of degree 1). Since D f ( v ) is also symmetric, this clearly implies that D f ( v ) = 0. The same argument applied to points of r.V , r >
0, shows that D f vanishes identically in the open connected cone C = X ∩ { rv : r > v ∈ V } .Therefore f ↾ C is the restriction to C of a linear function l : X → R , which in turnshows that V = C ∩ { l = 1 } is a line segment, in contradiction with the rotundityof f . This proves (A)We now turn to proving (B). Let V ⊆ G be an open interval containing v . With v ∈ V we associate a ( v ) = D f ( v )( e v , e v ), so that a ( v ) >
0. Since v a ( v ) iscontinuous, there is no restriction to assume that V is small enough for a ( v ) > a > v ∈ V . If t ∈ R then (42) implies k v + te v k = 1 + h te v , v ∗ i + 12 D f ( v )( te v , te v ) + o ( t ) = 1 + a ( v ) t + o ( t ) . We let h t ∈ X be so that v + h t ∈ S X , v + h t = v + te v k v + te v k , and we define ε t = k h t k . Since h t = (cid:18) t a ( v ) t + o ( t ) (cid:19) e v − (cid:18) a ( v ) t + o ( t )1 + a ( v ) t + o ( t ) (cid:19) v (43) we readily verify that lim t → ε t t = 1 uniformly in v ∈ V . (44)Furthermore, (42) shows that (cid:13)(cid:13)(cid:13)(cid:13) v + h t (cid:13)(cid:13)(cid:13)(cid:13) = 1 + 12 h h t , v ∗ i + 14 (cid:18) D f ( v )( h t , h t ) (cid:19) + o ( t ) , (45)and it follows from (43) that h h t , v ∗ i = t a ( v ) t + o ( t ) h e v , v ∗ i − a ( v ) t + o ( t )1 + a ( v ) t + o ( t ) h v, v ∗ i = − a ( v ) t + o ( t )1 + a ( v ) t + o ( t )= − a ( v ) t + o ( t ) , as well as12 D f ( v )( h t , h t ) = t (1 + a ( v ) t + o ( t )) (cid:18) D f ( v )( e v , e v ) (cid:19) + (cid:18) a ( v ) t + o ( t )1 + a ( v ) t + o ( t ) (cid:19) (cid:18) D f ( v )( v, v ) (cid:19) − (cid:18) t ( a ( v ) t + o ( t ))(1 + a ( v ) t + o ( t )) (cid:19) (cid:18) D f ( v )( v, e v ) (cid:19) = a ( v ) t (1 + a ( v ) t + o ( t )) + O ( t )= a ( v ) t + o ( t ) . Plugging these into (45) yields1 − (cid:13)(cid:13)(cid:13)(cid:13) v + h t (cid:13)(cid:13)(cid:13)(cid:13) = a ( v )4 t + o ( t ) . In view of (44) this means that one can find ε > − (cid:13)(cid:13)(cid:13)(cid:13) v + h t (cid:13)(cid:13)(cid:13)(cid:13) > a ( v )5 k h t k ∀k h t k ε . (46)Taking if necessary a smaller ε , we can also assume that any h ∈ X satisfying both v + h ∈ S X and k h k ε is of the form h t for some t .We let now ε > v + h ∈ S X with k h k > ε .We distinguish two cases. Firstly if k h k ε then (46) says1 − (cid:13)(cid:13)(cid:13)(cid:13) v + h (cid:13)(cid:13)(cid:13)(cid:13) > a ( v )5 ε . Secondly if k h k > ε then 1 − (cid:13)(cid:13)(cid:13)(cid:13) v + h (cid:13)(cid:13)(cid:13)(cid:13) > M ε , where M is defined by M := min ( − (cid:13)(cid:13) v + h (cid:13)(cid:13) k h k : { v, v + h } ∈ S X and k h k > ε ) , which is positive due to the uniform rotundity of the norm. We just have proved,owing also to Remark 2.11, that δ X ( v ; ε ) > C ε ∀ v ∈ V, with C = min( M, a ) and Conclusion (B) now easily follows. (cid:3) EIGHTED LENGTH 35 . —
Here we notice that 5.2 can be im-proved in the following way. In case x , x and z are the vertices of a nondegeneratetriangle in X , one has k x − z k + k z − x k > k x − x k (cid:18) δ X (cid:18) v ; dist( z, x + span { x − x } )2( k x − z k + k z − x k ) (cid:19)(cid:19) . where v = x − x k x − x k . The proof is exactly the same as that of 5.2 once one recognizes that δ X ( ε ) can bereplaced by δ X ( v ; ε ) in (14).7.5 (Height bound) . — Under our assumptions on X , the height bound 5.3(G)can be improved to dist( z, x + L ) .δ − X ( v ; ξ ( r ))where v = ( x − x ) k x − x k − . Here δ − X ( v ; ξ ( r )) denotes the value at ξ ( r ) of thereciprocal of the function δ X ( v ; · ). The proof, based on 7.4, is identical.7.6 (Tangent lines) . — We now indicate how to apply the previous “localized”observations to the study of regularity of almost minimizing curves, using the notionof tangent measure . We point out that everything we do in this number makes sensein a finite dimensional rotund Banach space. We will often refer to [6] and [15], butonly to elementary results in these papers, in particular those that do not dependon the Euclidean structure of the ambient space R n considered there.Given C ⊆ X a set which is ( ξ, r ) almost minimizing in U ⊆ X with respect to aDini gauge ξ , we consider the finite Borel measure µ = H C in U . Given x ∈ U and r > r − ( T x,r ) ∗ µ on ( U − x ) /r , where T x,r ( y ) = ( y − x ) /r . A 1 dimensional tangent measure of µ at x is, by definition, aweak* limit of some sequence { r − j ( T x,r j ) ∗ µ } , where r j ↓
0. The collection of thoseis denoted Tan (1) ( µ, x ). We gather useful information about tangent measures.(A) If x ∈ C ∩ U then Tan (1) ( µ, x ) = ∅ ;(B) If x ∈ C ∩ U , Θ ( H C, x ) = 1, and ν ∈ Tan (1) ( µ, x ), then ν = H W for some line W ∈ G ( X, x ∈ C ∩ U and Θ ( H C, x ) = 1, recall 4.11 that x is a regular point of C : If γ is an arclength parametrization of some C ∩ B ( x, r ) with γ (0) = x ,then γ is differentiable at 0 if and only if Tan (1) ( µ, x ) is singletonic;(D) If x ∈ C ∩ U , Θ ( H C, x ) = 1, then the collection of tangent lines G ( X, ∩ { W : H W ∈ Tan (1) ( µ, x ) } is connected. Proof of (A) . The proof of (A) depends on our assumption dim
X < ∞ throughthe application of a compactness Theorem for Radon measures, see e.g. [6, Propo-sition 5.4]. The relevant almost monotonicity property of µ is in 4.7. (cid:3) Proof of (B) . Let r j ↓ ν is the weak* limit of the sequence { µ j } where µ j = r − j ( T x,r j ) ∗ µ . Abbreviate C j = ( C − x ) /r j . Using the translation invariance,and behavior under homothethy, of the Hausdorff measure, one immediately checksthat C j is ( ξ j , r /r j ) almost minimizing in U x,r j = ( U − x ) /r j , where ξ j ( t ) = ξ ( tr j ),and that µ j = H C j . In the vocabulary of [6], each µ j is 1 concentrated,according to 4.6(C). Notice that 4.7 does not show µ j is ( ξ j ,
1) almost monotone inthe sense of [6] (because the condition is verified only at those points of the supportof µ j , not all points of U x,r j ): We will emphasize this by saying that µ j is ( ξ j , on its support . Carefully reading the proofs of [6, 4.2 and 4.1]reveals that (1) For every λ > δ > j such that C j ∩ B (0 , λ ) ⊆ B (supp ν, δ ) for every j > j ;(2) Θ ( ν, z ) > z ∈ supp ν .We put Z = supp ν , Z λ = Z ∩ B (0 , λ ) and F λ = Z λ ∩ bdry B (0 , λ ) for each λ > { j k } satisfying P k k ξ (2 k r j k ) < /
6. By the assumption onthe density and 4.4 (D), we may assume that H ( C j k ∩ B (0 , k ))2 k +1 − < , and that card C j k ∩ bdry B (0 , k ) > k >
1. Setting G k := { λ ∈ [0 , k ] : card C j k ∩ bdry B (0 , λ ) > } , we infer from 2.7 that L ( G k ) < k / k > λ k ∈ (2 k +1 / , k ) such that card C j k ∩ bdry B (0 , λ k ) = 2. Writing { x k , x k } := C j k ∩ bdry B (0 , λ k ), the set C ′ j k := ( C j k \ B (0 , λ k )) ∪ [0 , x k ] ∪ [0 , x k ] is a competitorfor C j k . From the almost minimality of C j k we infer that H ( C j k ∩ B (0 , λ k )) λ k (1 + ξ ( λ k r j k )) k +1 + 2 k +1 ξ (2 k r j k ) . Setting H k := G k ∩ [0 , λ k ], we deduce from 2.7 that L ( H k ) k +1 ξ (2 k r j k ).Therefore L ( H ) < / H := ∪ k H k . Consequently, for each integer k ≥ R k ∈ ( λ k − / , λ k ) \ G . We shall keep in mind that R k ∈ (2 k − , k )by construction.Let us now fix an arbitrary integer k >
1. The way we have selected the radius R k ensures that card C j h ∩ bdry B (0 , R k ) = 2 for every h > k . We write { y k,h , y k,h } := C j h ∩ bdry B (0 , R k ). Noticing that the set C ′′ j h := ( C j h \ B (0 , R k )) ∪ [ y k,h , y k,h ] isa competitor for C j h , the almost minimality of C j h yields H ( C j h ∩ B (0 , R k )) (1 + ξ ( R k r j h )) k y k,h − y k,h k . On the other hand, lim h H ( C j h ∩ B (0 , R k )) = 2 R k = diam B (0 , R k ) by the densityassumption. By the inequality above, it then follows that lim h k y k,h − y k,h k = 2 R k .Referring to (1), we now choose a sequence δ h ↓ h > k there exist z k,h , z k,h ∈ B (0 , R k + δ h ) with k y ik,h − z ik,h k δ h , i = 1 ,
2. Taking asubsequence if necessary, z ik,h → z ik as h → ∞ , z ik ∈ F R k , and k z k − z k k = 2 R k .According to 5.2, the rotundity of the norm implies that the line segment [ z k , z k ]is a diameter of B (0 , R k ), i.e. 0 ∈ [ z k , z k ]. We claim that [ z k , z k ] ⊆ Z R k . To provethe claim, we first notice that C j h ∩ B (0 , R k ) contains a curve Γ h whose endpointsare { y k,h , y k,h } , see 4.4 (E). By the Blashke Selection Principle, up to a furthersubsequence, Γ h → K ∗ in the Hausdorff distance for some compact connectedset K ∗ ⊆ B (0 , R k ) containing z k and z k . In particular H ( K ∗ ) > k z k − z k k =2 R k . On the other hand, by lower semicontinuity of H with respect to Hausdorffconvergence, we have H ( K ∗ ) lim inf h →∞ H (Γ h ) lim h →∞ H ( C j h ∩ B (0 , R k )) = 2 R k . Hence H ( K ∗ ) = 2 R k , and the rotundity of the norm yields K ∗ = [ z k , z k ]. In viewof (1), we have thus proved the claim. It now follows from (2) and [8, 2.10.19(3)]that ν B (0 , R k ) > H [ z k , z k ]. Finally, the monotonicity stated in 4.7 andthe density assumption classicaly implies that ν ( B (0 , λ )) = 2 λ for every λ > ν ( B (0 , R k )) = H ([ z k , z k ]), whence ν B (0 , R k ) = H [ z k , z k ]. Fromthe arbitrariness of k we conclude that Z is a line through the origin and ν = H Z , which completes the proof of (B). (cid:3) EIGHTED LENGTH 37
We leave the easy proof of (C) to the reader. It relies on the local convergenceof C j to Z in Hausdorff distance, as was already used in the proof of (B).Conclusion (D) is our principal tool in this section. It is a consequence of [15,Theorem 2.6] as we now explain. Proof of (D) . Let us recall that a tangent measure of µ at the point x in the senseof [15] is a weak* limit of some sequence { c j ( T x,r j ) ∗ µ } , where r j ↓ c j > µ, x ) the collection of all tangentmeasures of µ at x . The set Tan( µ, x ) is endowed with the topology induced by themetric D ( ν , ν ) := X p ∈ N − p min(1 , F p ( ν , ν )) , where F p ( ν , ν ) := sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z R n f dν − Z R n f dν (cid:12)(cid:12)(cid:12)(cid:12) : supp f ⊆ B (0 , p ) , f > , Lip( f ) (cid:27) (recall that the weak* convergence of Radon measures coincides with the conver-gence with respect to D ). Now we observe that 3.2 together with 4.7 shows thatfor r > κ − r µ ( B ( x, r )) r κr , for some constant κ >
0. It implies that Tan( µ, x ) = { cν : c > , ν ∈ Tan (1) ( µ, x ) } .In turn, we infer from (B) that Tan( µ, x ) = { c H W : c > , W ∈ G ( X, } .According to [15, Theorem 2.6], we can conclude that Tan( µ, x ) is connected. It isnow elementary to check that the map ν = c H W ∈ Tan( µ, x ) W ∈ G ( X, (cid:3) Theorem (Differentiable regularity) . —
We recall our general assumptionfor this section that X is a 2 dimensional rotund Banach space. Assume that (A) C ⊆ X is compact, connected, U ⊆ X is open, x ∈ C ∩ U , r > , B ( x , r ) ⊆ U ; (B) ξ is a gauge and √ ξ is Dini; (C) C is ( ξ, r ) almost minimizing in U ; (D) Θ ( H C, x ) = 1 .It follows that there exists r > such that C ∩ B ( x , r ) is a differentiable curve.Proof. We explain how to modify the proof of 5.5. Since x is a regular point of C ,recall 4.10, we may assume (letting r be smaller if necessary) each x ∈ C ∩ B ( x , r )is regular as well, and C ∩ B ( x, r ) is a Lipschitz curve, according to 4.11. In orderto establish the differentiability of a corresponding arclength parametrization, itsuffices to show Tan (1) ( µ, x ) contains excatly one element, according to 7.6(C).We let G be the set associated with the norm of X in 7.3. Abbreviate Tan( C, x ) = S X ∩ { v : H span { v } ∈ Tan (1) ( H C, x ) } . We will consider the followingalternative: Either Tan( C, x ) ∩ G = ∅ , or Tan(
C, x ) ∩ G = ∅ . In the first case,the connectedness property stated in 7.6(D) and the fact that S X \ G has emptyinterior imply easily that Tan( C, x ) is a singleton and the proof is complete. Inthe second case we pick span { v } = W ∈ G ( X,
1) such that v ∈ G , H W isa tangent measure to H C at x , and we choose η > v ∈ S X and k v − v k < η then v ∈ G . The proof of 5.5 remains unchanged until the end ofthe proof of Claim . The application of 5.3 is replaced by an application of 7.5together with 7.3(B). To each r > x , r ) is a good pair we associatethe unit vector v r generating the line that joins the endpoints of Γ x ,r . We maychoose r > k v r − v k < η/
2. We proceed through the remainingpart of the proof of 5.5 until near (39). In our new notations, this shows that k v r j +1 − v r j k C p ξ ( r j ). However for the computations to be valid, one mustmake sure at each stage of the iteration that the unit vectors v r j still belong to G ,in fact we ought to guarantee that k v r j − v k < η in order that 7.3(B) applies. Thisof course can be enforced by choosing r so small that C P ∞ j =1 √ r j < η/ (cid:3) References
1. F.J. Almgren,
Existence and regularity almost everywhere of solutions to elliptic variationalproblems with constraints , Memoirs of the AMS, no. 165, American Math. Soc., 1976.2. L. Ambrosio and B. Kirchheim,
Currents in metric spaces , Acta Math. (2000), no. 1,1–80.3. Luigi Ambrosio and Paolo Tilli,
Topics on analysis in metric spaces , Oxford Lecture Se-ries in Mathematics and its Applications, vol. 25, Oxford University Press, Oxford, 2004.MR 2039660 (2004k:28001)4. D.L. Cohn,
Measure theory , Birkh¨auser, 1980.5. Guy David,
H¨older regularity of two-dimensional almost-minimal sets in R n , Ann. Fac. Sci.Toulouse Math. (6) (2009), no. 1, 65–246. MR MR25181046. Th. De Pauw, Nearly flat almost monotone measures are big pieces of Lipschitz graphs , J. ofGeom. Anal. (2002), no. 1, 29–61.7. Robert Deville, Gilles Godefroy, and V´aclav Zizler, Smoothness and renormings in Banachspaces , Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, LongmanScientific & Technical, Harlow, 1993. MR 1211634 (94d:46012)8. Herbert Federer,
Geometric Measure Theory , Die grundlehren der mathematischen wis-senschaften, vol. 153, Springer-Verlag, New York, 1969.9. Mikhael Gromov,
Filling Riemannian manifolds , J. Differential Geom. (1983), no. 1, 1–147. MR 697984 (85h:53029)10. Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorffmeasure , Proc. Amer. Math. Soc. (1994), no. 1, 113–123.11. Joram Lindenstrauss and Lior Tzafriri,
Classical Banach spaces. II. Function spaces , Ergeb-nisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas],vol. 97, Springer-Verlag, Berlin, 1979, Function spaces. MR 540367 (81c:46001)12. Gaven J. Martin,
Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains andthe quasihyperbolic metric , Trans. Amer. Math. Soc. (1985), no. 1, 169–191. MR 805959(87a:30037)13. Thomas Meinguet, ( M , cr γ , δ ) -minimizing curve regularity , Bull. Belg. Math. Soc. SimonStevin (2009), no. 4, 577–591. MR 2583547 (2011a:49093)14. F. Morgan, ( M , ε, δ ) -minimal curve regularity , Proc. Amer. Math. Soc. (1994), 677–686.15. D. Preiss, Geometry of measures in R n : Distribution, rectifiability, and densities , Ann. ofMath. (2) (1987), 537–643.16. Jussi V¨ais¨al¨a, Quasihyperbolic geodesics in convex domains , Results Math. (2005), no. 1-2,184–195. MR 2181248 (2006h:30035)17. , Quasihyperbolic geometry of domains in Hilbert spaces , Ann. Acad. Sci. Fenn. Math. (2007), no. 2, 559–578. MR 2337495 (2008d:30041) E-mail address : [email protected] E-mail address : [email protected] E-mail address ::