A geometric second-order-rectifiable stratification for closed subsets of Euclidean space
aa r X i v : . [ m a t h . C A ] A p r A geometric second-order-rectifiable stratificationfor closed subsets of Euclidean space
Ulrich Menne Mario SantilliOctober 17, 2018
Abstract
Defining the m -th stratum of a closed subset of an n dimensionalEuclidean space to consist of those points, where it can be touched bya ball from at least n − m linearly independent directions, we establishthat the m -th stratum is second-order rectifiable of dimension m and aBorel set. This was known for convex sets, but is new even for sets ofpositive reach. The result is based on a new criterion for second-orderrectifiability. MSC-classes 2010.
Keywords.
Second-order rectifiability, distance bundle, normal bundle, coareaformula, stratification.
The main purpose of the present paper is to establish the following theorem;our notation is based on [Fed69, pp. 669–676], see the end of this introduction.
Structural theorem on the singularities of closed sets (see 4 . . Sup-pose A is a closed subset of R n , for a ∈ A , Dis(
A, a ) is the set of v ∈ R n satisfying A ∩ { x : | x − ( a + v ) | < | v |} = ∅ , m is an integer, ≤ m ≤ n , and B = A ∩ { a : Dis( A, a ) contains at least n − m linearly independent vectors } . Then, B can be H m almost covered by the union of a countable collectionof m dimensional, twice continuously differentiable submanifolds of R n . In the terminology of [San17, p. 2] for m ≥
1, the conclusion asserts that B is countably ( H m , m ) rectifiable of class 2. If A is convex, then B consistsof the set of points, where the dimension of the normal cone of A is at least n − m , see 4.14. Hence, our theorem contains Alberti’s structural theorem onthe singularities of convex sets, see [Alb94, Theorem 3]. We also prove, that B is a countably m rectifiable Borel set, see 4.12; in particular, if m ≥
1, then B can be covered (without exceptional set) by a countable family of images ofLipschitzian functions from R m into R n , and, if m = 0, then B is countable.Our approach rests on two pillars. The first may be stated as follows.1 arametric criterion for second-order rectifiability (see 2 . . Suppose W is an L n measurable subset of R n , m is an integer, ≤ m ≤ n , f : W → R ν is a locally Lipschitzian map, Z = R ν ∩ { z : H n − m ( f − [ { z } ]) > } , and, for H m almost all z ∈ Z , there exists an m dimensional subspace U of R ν satisfying lim sup y → x | y − x | − dist( f ( y ) − f ( x ) , U ) < ∞ whenever x ∈ f − [ { z } ] . Then, Z can be H m almost covered by the union of a countable collectionof m dimensional, twice continuously differentiable submanifolds of R ν . Notice that f − [ { z } ] abbreviates { x : f ( x ) = z } , see below. The key toreduce this criterion to the nonparametric case is the construction (in 2.1) of acountable collection G of m rectifiable subsets P of W with H m ( Z ∼ f [ S G ]) =0 such that, for each P ∈ G , the restriction f | P is univalent and ( f | P ) − isLipschitzian. The nonparametric case was comprehensively studied in [San17];however, for the present purpose, also [Sch09] would be sufficient (see 2.6).The second pillar of the proof of the structural theorem is the next resultthat we state here for the special case of a convex set A . It concerns the relationof the nearest point projection, ξ A , with the tangent and normal cones of A . A geometric observation for convex sets (see 4 .
11 with 3 . , . . If A is a nonempty closed convex subset of R n , m is an integer, ≤ m < n , x ∈ R n ∼ A , a = ξ A ( x ) , dim Nor( A, a ) ≥ n − m , U is an m dimensional subspaceof R n , U ⊂ Tan(
A, a ) , and x − a belongs the relative interior of Nor(
A, a ) , then lim sup y → x | y − x | − dist( ξ A ( y ) − a, U ) < ∞ . This criterion and its generalisation to closed sets in 4.11 owe much to Fed-erer’s treatment of sets of positive reach (a concept that embraces convex setsand submanifolds of class 2) in [Fed59]. Since it is elementary, that the set B inthe structural theorem is countably m rectifiable, the parametric criterion forsecond-order rectifiability then is readily applied with f = ξ A | W for suitable W . Connection to curvature measures
Instead of using second-order rectifi-ability properties, curvature properties can also be studied via general Steinerformulae. This approach was taken, for sets of positive reach and various moregeneral classes of sets, by Federer in [Fed59], Stach´o in [Sta79], Z¨ahle in [Z¨ah86],Rataj and Z¨ahle in [RZ01], and Hug, Last, and Weil in [HLW04]; in fact, [Sta79]and [HLW04] treat arbitrary closed subsets of Euclidean space. Accordingly,the natural question (under investigation by the second author) arises to char-acterise the relation of both notions of curvature.
Connection to varifold theory
The original motivation of the first authorfor the present study was to create a deeper understanding of a relation provenby Almgren in his area-mean-curvature characterisation of the sphere in [Alm86].There, an equation relating the curvature measures (similar to those of [Z¨ah86])of the convex hull of the support of a certain varifold to the perpendicular partof the mean curvature of the varifold is established in [Alm86, § cknowledgements The first author thanks the participants of an onlinereading seminar of [Alm86] for their early interest in these developments. Thematerial of this paper originates from the PhD thesis of the second author,supervised by the first author, submitted at the University of Potsdam. Thepaper was written while both authors worked at the Max Planck Institute forGravitational Physics (Albert Einstein Institute) and the University of Potsdam.
Notation
Our notation and terminology is that of [Fed69, pp. 669–676], ex-cept that, as in [Kel75, p. 8], we denote the image of A under a relation r by r [ A ] = { y : ( x, y ) ∈ r for some x ∈ A } . The purpose of the present section is to prove the parametric criterion for second-order rectifiability in 2.5. We begin by establishing a theorem that allows toconstruct univalent parametrisations from a Lipschitzian given one.
Suppose W is an L n measurable subset of R n , m is an integer, ≤ m ≤ n , and f : W → R ν is a locally Lipschitzian map. Then, there existsa countable collection G of compact subsets P of W , such that f | P is univalentand ( f | P ) − is Lipschitzian, satisfying H m (cid:0) R ν ∩ { z : H n − m ( f − [ { z } ]) > } ∼ S { f [ P ] : P ∈ G } (cid:1) = 0 . Moreover, each member of G is contained in some m dimensional affine plane.Proof. We firstly consider the special case that W is a compact subset of R n .Choose F : R n → R ν with F | W = f and Lip F = Lip f < ∞ by Kirszbraun’stheorem [Fed69, 2.10.43]; in particular, D F is a Borel function whose domain isa Borel set by [Fed69, 3.1.2]. Defining Z i = R ν ∩ { z : H n − m ( f − [ { z } ]) ≥ /i } whenever i is a positive integer, we note that R ν ∩ { z : H n − m ( f − [ { z } ]) > } = S ∞ i =1 Z i . Moreover, the sets Z i are Borel sets by [Fed69, 2.10.26] and ( H m , m ) rectifiableby [Fed69, 3.2.31]. We define, for every positive integer i , the class Ω i to consistof all families G of compact subsets P of f − [ Z i ] such that f [ P ] ∩ f [ Q ] = ∅ if and only if P = Q whenever P, Q ∈ G , and such that H m ( P ) > , f | P is univalent , ( f | P ) − is Lipschitzian ,P is contained in some m dimensional affine subspace of R n whenever P ∈ G . Clearly, each member of Ω i is countable. Using Hausdorff’smaximal principle (see [Kel75, p. 33]), we choose maximal elements G i of Ω i .The proof of the present case will be concluded by establishing H m (cid:0) Z i ∼ S { f [ P ] : P ∈ G i } (cid:1) = 0 for every positive integer i. i and define Borel sets T = Z i ∼ S { f [ P ] : P ∈ G i } and S = f − [ T ]. If T had positive H m measure, then, noting [Fed69, 2.10.35], B = S ∩ (cid:8) w : k V m D F ( w ) k > (cid:9) would be a Borel set and have positive L n measure by the coarea formula[Fed69, 3.2.22 (3)] with W , Z , and f replaced by S , T , and f | S , since( L n x S, n ) ap D( f | S )( w ) = D F ( w ) for L n almost all w ∈ S by [Fed69, 2.10.19 (4)].Consequently, identifying R n ≃ R m × R n − m , there would exist a linearisometry g : R n → R n such that L n ( A ) > A = B ∩ (cid:8) w : V m (D F ( w ) | g [ R m × { } ]) = 0 (cid:9) = g (cid:2) g − [ B ] ∩ { x : V m (D( F ◦ g )( x ) | R m × { } ) = 0 } (cid:3) and, as A would be a Borel set, η ∈ R n − m so that L m ( R ) > R = R m ∩ { ξ :( ξ, η ) ∈ g − [ A ] } by Fubini’s theorem, see [Fed69, 2.6.2 (3)]. Since R would be a Borel set, wecould apply [Fed69, 3.2.2] to the function h : R m → R ν defined by h ( ξ ) =( F ◦ g )( ξ, η ) for ξ ∈ R m , and use the Borel regularity of H m to construct asubset P of g [ R × { η } ] with G i ∪ { P } ∈ Ω i , contrary to the maximality of G i .To treat the general case , we choose an increasing sequence of compact sub-sets K i of R n with L n (cid:0) W ∼ S ∞ i =1 K i (cid:1) = 0. Since, in conjunction with [Fed69,2.4.5], [Fed69, 2.10.25] applied with A replaced by W ∼ S ∞ i =1 K i implies that H n − m (cid:0) f − [ { z } ] ∼ S ∞ i =1 K i (cid:1) = 0 for H m almost all z ∈ R ν and lim i →∞ H n − m ( f − [ { z } ] ∩ K i ) = H n − m ( f − [ { z } ]) for such z , we readilyinfer the conclusion. The contradiction argument is inspired by [Fed69, 3.2.21].
For the nearest point projection onto a set of positive reach, theidea of exhaustion by means of images from lower dimensional parts of thedomain of f is employed in [Fed59, 4.15 (3)]. The important additional featureof members P in our collection G is the Lipschitz continuity of ( f | P ) − . One readily verifies that 2.1 also holds with m = 0, but this willnot be needed in the present paper.The parametric criterion for second-order rectifiability now reads as follows. Under the hypotheses of 2.1, if Z = R ν ∩ { z : H n − m ( f − [ { z } ]) > } , and, for H m almost all z ∈ Z , there exists an m dimensional subspace U of R ν satisfying lim sup y → x | y − x | − dist( f ( y ) − f ( x ) , U ) < ∞ whenever x ∈ f − [ { z } ] , then Z can be H m almost covered by the union of a countable collection of m dimensional submanifolds of R ν of class . roof. Whenever P ∈ G , as ( f | P ) − is Lipschitzian, we notice that, for H m almost all z ∈ Z ∩ f [ P ], there exists an m dimensional subspace U of R ν suchthat lim sup f [ P ] ∋ ζ → z | ζ − z | − dist( ζ − z, U ) < ∞ . Therefore, the conclusion follows from [San17, 5.3] and [Fed69, 3.1.15].
With little additional effort, the final argument could have beenbased on [Sch09, A.1] instead of [San17, 5.3] and [Fed69, 3.1.15].
In conjunction with the preceding corollary, the following observa-tion will be useful. If B is a countably ( H m , m ) rectifiable subset of R ν , then,for H m almost all b ∈ B , there exists an m dimensional subspace U of R ν suchthat U ⊂ Tan(
B, b ) ; in fact, [Fed69, 2.1.4, 3.1.21] reduce the problem to Borelsets B , in which case [Fed69, 2.10.19 (4), 3.2.17, 3.2.18] apply. In the present section, we mainly collect some basic properties of convex setsand related definitions in 3.1–3.10 for convenient reference. Additionally, wenote an observation concerning convex cones in 3.12–3.14.
Suppose A ⊂ R n and x ∈ R n . Then, the distance of x to A isdenoted by dist( x, A ) = inf {| x − a | : a ∈ A } . If A = ∅ , then dist( · , A ) is real valued and Lip dist( · , A ) ≤ If R = ( R n × A ) ∩ { ( x, a ) : | x − a | = dist( x, A ) } , then, using 3.2,one verifies that { a : ( x, a ) ∈ R for some x ∈ B } is bounded whenever B is abounded subset of R n . Moreover, if A is closed, so is R . (see [Fed59, 4.1]) . Suppose A ⊂ R n and U is the set of all x ∈ R n such that there exists a unique a ∈ A with | x − a | = dist( x, A ). Then,the nearest point projection onto A is the map ξ A : U → A characterised by therequirement | x − ξ A ( x ) | = dist( x, A ) for x ∈ U . Using 3.3, we obtain that the function ξ A is continuous. Moreover,if A is closed, then dmn ξ A is a Borel set; in fact, one verifies, by means of 3.3,that the function mapping x ∈ R n onto d ( x ) = diam { a :( x, a ) ∈ R } ∈ R isupper semicontinuous, and dmn ξ A = { x : d ( x ) = 0 } . (see [Sch14, p. xix]) . If A ⊂ R n , then aff A denotes the affinehull of A . (see [Sch14, p. 7, p. xx]) . Suppose C is a convex subset of R n .Then, the dimension of C , denoted by dim C , is defined to be the dimension ofaff C , and the relative boundary [ interior ] of C is defined to be the boundary[interior] of C relative to aff C . If V is the relative interior of C , then V is convex, dim V = dim C ,and c + t ( v − c ) ∈ V whenever v ∈ V , c ∈ C , and 0 < t ≤ C = R n , this is [Sch14, 1.1.9, 1.1.10, 1.1.13].5 .9 Lemma. Suppose C is a nonempty closed convex subset of R n .Then, the following four statements hold.(1) There holds dmn ξ C = R n and Lip ξ C ≤ .(2) If c ∈ C , then Tan(
C, c ) = R n ∩ { u : u • v ≤ v ∈ Nor(
C, c ) } and C ⊂ { c + u : u ∈ Tan(
C, c ) } ⊂ aff C ; in particular, dim C = dim Tan( C, c ) .(3) If c ∈ C , then Nor(
C, c ) = { v : ξ C ( c + v ) = c } = R n ∩ { v : v • ( x − c ) ≤ x ∈ C } . (4) If B is the relative boundary of C , then B = C ∩ { c : c + v ∈ aff C for some v ∈ S n − ∩ Nor(
C, c ) } . Proof. (1) is asserted in [Fed69, 4.1.16]. In view of (1), the first equation andthe first inclusion in (2) are contained in [Fed59, 4.8 (12)] and [Fed59, 4.18],respectively; the remaining items of (2) then follow. The first equation in (3)follows from (1) and [Fed59, 4.8 (12)]. The second equation in (3) follows from[KS80, I.2.3]. Finally, (4) is implied by [Sch14, 1.3.2].
Suppose X = R n ∩ B (0 , , F is the family of nonempty closedsubsets of X endowed with the Hausdorff metric, and G = F ∩{ C : C is convex } .Then, the following four statements hold.(1) The families F and G are compact.(2) The function mapping ( x, B ) ∈ X × F onto dist( x, B ) ∈ R is continuous.(3) The function mapping C ∈ G onto dim C ∈ Z is lower semicontinuous.(4) If Φ = ( G × F ) ∩ { ( C, B ) : B is the relative boundary of C } , then Φ is aBorel function whose domain equals the Borel set G ∩ { C : dim C ≥ } .Proof. (1) is contained in [Fed69, 2.10.21]. (2) follows from 3.2. We observe that,in order to prove (3) and (4), it sufficient to establish the following assertion. If k is an integer, C i is a sequence in G with dim C i = k , C ∈ G , and C i → C as i → ∞ , then dim C ≤ k and, in case of equality with k ≥ , also Φ( C ) =lim i →∞ Φ( C i ) . For this purpose, we assume, possibly passing to a subsequence,that for some affine subspace Q of R n dist( v, aff C i ) → dist( v, Q ) as i → ∞ for v ∈ R n , and, if k ≥
1, that for some B ∈ F , we have Φ( C i ) → B as i → ∞ . It follows C ⊂ Q , whence we infer dim C ≤ dim Q ≤ k . Therefore, if dim C = k ≥
1, then Q = aff C and we could assume, possibly replacing C i by g − i [ C i ] for a sequenceof isometries g i of R n with lim i →∞ g i ( x ) = x for x ∈ R n and using 3.2, that C i ⊂ Q for each index i ; in which case Φ( C ) = B follows readily from 3.9 (4). We observe that (2)–(4) imply that, if A is a Borel subset of R n and Γ : A → G is a Borel function, then the set of ( a, v ) ∈ A × R n such that v belongs to the relative interior of Γ( a ) is a Borel subset of R n × R n . A subset C of R n is said to be a cone if and only if λc ∈ C whenever 0 < λ < ∞ and c ∈ C . Suppose C is a convex cone in R n , D = R n ∩ { d : d • c ≤ c ∈ C } ,U is an m dimensional plane in R n , U ⊂ D , dim C ≥ n − m , and v belongs tothe relative interior of C .Then, dim C = n − m and there exists ≤ γ < ∞ satisfying dist( d, U ) ≤ − γd • v for d ∈ D. Proof.
Defining V = R n ∩ { v : u • v = 0 for u ∈ U } , we see C ⊂ V from [Fed59,4.5], hence aff C = V ; in particular, dim C = n − m . Since D is closed underaddition and U ⊂ D , D is invariant under directions in U . Therefore, it issufficient to prove the existence of 0 ≤ γ < ∞ such that the inequality holds for d ∈ D ∩ V ∩ S n − . If there were no such γ , then, by compactness, there wouldexist d ∈ D ∩ V ∩ S n − with d • v = 0 which would imply that v belongs to therelative boundary of C , as d ∈ aff C . Under the hypotheses of 3.13, there holds dist( b, U ) ≤ − γb • v + (1 + γ | v | ) dist( b, D ) for b ∈ R n . Proof.
In view of 3.2 and 3.9 (1), one may apply 3.13 to d = ξ D ( b ). In the present section, we introduce the distance bundle in 4.1–4.6; its nonzerodirections correspond to the normal bundle employed by Hug, Last, and Weilin [HLW04], see 4.6. Then, we extend (in 4.7) some basic estimates from Fed-erer’s treatment of sets of positive reach in [Fed59] which lead to an importantone-sided estimate for the nearest point projection in 4.9. Finally, we derive thegeometric observation, described for convex sets in the introduction, in 4.11,and the main structural theorem on the singularities of closed sets in 4.12.
Suppose A ⊂ R n . Then, the distance bundle of A is definedby Dis( A ) = ( R n × R n ) ∩ { ( a, v ) : a ∈ Clos A and | v | = dist( a + v, A ) } . Moreover, we let Dis(
A, a ) = { v :( a, v ) ∈ Dis( A ) } for a ∈ R n . Clearly, Dis( A ) = Dis(Clos( A )), Dis( A ) is closed, and 0 ∈ Dis(
A, a )if and only if a ∈ Clos A . Moreover, Dis( A, a ) is a convex subset of Nor(
A, a )for a ∈ R n by [Fed59, 4.8 (2)]. If X and G are as in 3.10, then the function mapping a ∈ Clos A onto Dis( A, a ) ∩ X ∈ G is a Borel function; in fact, 4.2 implies that, in theterminology of [CV77, II.20], the function in question is an upper semicontinuousmultifunction, whence the assertion follows by [CV77, III.3].7 .4 Remark. If a ∈ A , v ∈ Dis(
A, a ), and 0 ≤ t <
1, then ξ A ( a + tv ) = a . Inparticular, ξ A ( a + v ) = a whenever v belongs to the relative interior of Dis( A, a ),and Dis(
A, a ) is the closure of { v : ξ A ( a + v ) = a } . In view of 4.4, we could have alternatively formulated our maintheorem (see 4.12), for closed sets, in terms of the bundle { ( a, v ) : ξ A ( a + v ) = a } which would be more in line with Stach´o’s definition of prenormals in [Sta79,p. 192]. Our choice of bundle is motivated by the fact that Dis( A ) is closed. If A is closed, then 4.4 yields that (cid:8) ( a, | v | − v ) : ( a, v ) ∈ R n × R n and 0 = v ∈ Dis(
A, a ) (cid:9) equals the normal bundle of A defined in [HLW04, p. 239].Basic estimates for the distance bundle are collected in the following theorem. Suppose A ⊂ R n . Then, the following three statements hold.(1) If < q < ∞ , a ∈ Clos A , b ∈ Clos A , v ∈ R n , andeither v = 0 or q | v | − v ∈ Dis(
A, a ) , then ( b − a ) • v ≤ (2 q ) − | b − a | | v | .(2) If < r < q < ∞ , x ∈ R n , y ∈ R n , a ∈ A , b ∈ A , and | x − a | = dist( x, A ) ≤ r, | y − b | = dist( y, A ) ≤ r, either x = a or q | x − a | − ( x − a ) ∈ Dis(
A, a ) , either y = b or q | y − b | − ( y − b ) ∈ Dis(
A, b ) , then ξ A ( x ) = a , ξ A ( y ) = b , and | b − a | ≤ q ( q − r ) − | y − x | . (3) If < q < ∞ , a ∈ Clos A , b ∈ Clos A , C is a convex cone in R n , qv ∈ Dis(
A, a ) whenever v ∈ C ∩ S n − , and D = R n ∩ { u : u • v ≤ v ∈ C } , then dist( b − a, D ) ≤ (2 q ) − | b − a | . Proof.
To prove (1), we assume v = 0, let w = | v | − v , and compute | a + qw − b | ≥ dist( a + qw, A ) = q, | a − b | + 2 qw • ( a − b ) + q ≥ q , qw • ( b − a ) ≤ | b − a | , v • ( b − a ) ≤ (2 q ) − | b − a | | v | . To prove (2), we notice that a = ξ A ( x ) and b = ξ A ( y ) by 4.4 and infer( b − a ) • ( x − a ) ≤ | b − a | r/ (2 q ) , ( a − b ) • ( y − b ) ≤ | b − a | r/ (2 q ) . from applying (1) twice; once with v replaced by x − a and once with a , b , and v replaced by b , a , and y − b . Therefore, we obtain | b − a || y − x | ≥ ( b − a ) • ( y − x )= ( b − a ) • (( b − a ) + ( a − x ) + ( y − b )) ≥ | b − a | (1 − r/q ) , | x − y | ≥ | a − b | ( q − r ) /q .To prove (3), we suppose a = 0. Whenever v ∈ C , we notice that v • b ≤ (2 q ) − | b | | v | by (1), and estimate | b − v | = | b | + | v | − b • v ≥ | b | + | v | − | b | | v | /q ≥ | b | − | b | / (4 q ) . Consequently, dist( b, C ) ≥ | b | − | b | / (4 q ) and (3) is implied by [Fed59, 4.16]. The proof is almost verbatim taken from [Fed59, 4.8 (7) (8), 4.18 (2)],where slightly stronger hypotheses were made.Next, we derive a crucial one-sided estimate for the nearest point projection.
Suppose A ⊂ R n , < s < r < q < ∞ , and x ∈ dmn ξ A , s ≤ dist( x, A ) ≤ r, v = x − ξ A ( x ) | x − ξ A ( x ) | , qv ∈ Dis( A, ξ A ( x )) ,y ∈ dmn ξ A , s ≤ dist( y, A ) ≤ r, w = y − ξ A ( y ) | y − ξ A ( y ) | , qw ∈ Dis( A, ξ A ( y )) . Then, there holds ( ξ A ( x ) − ξ A ( y )) • v ≤ κ | y − x | , where κ = (2 s ) − (1 + 2 q/ ( q − r )) .Proof. We let a = ξ A ( x ) and b = ξ A ( y ), hence a = x −| x − a | v and b = y −| y − b | w .Next, we estimate ( a − b ) • v ≤ (2 s ) − | y − x | in case dist( x, A ) = dist( y, A ) = s ; in fact, noting dist( y, A ) ≤ | y − a | and | v | = | w | = 1, we obtain s ≤ | y − ( x − sv ) | , ( x − y ) • v ≤ (2 s ) − | y − x | , ( w − v ) • v ≤ , ( a − b ) • v = ( x − y ) • v + s ( w − v ) • v ≤ (2 s ) − | y − x | . In the general case, we let (see 3.9 (1)) α = a + ξ B (0 ,s ) ( x − a ) , β = b + ξ B (0 ,s ) ( y − b ) , notice α = a + sv and β = b + sw , and infer α ∈ dmn ξ A , ξ A ( α ) = a, β ∈ dmn ξ A , ξ A ( β ) = b, | β − α | ≤ | y − x | + 2 | b − a | ≤ (1 + 2 q/ ( q − r )) | y − x | from 4.4, 3.9 (1), and 4.7 (2). Therefore, we may apply the previous case with x and y replaced by α and β to deduce the conclusion. One could also derive a two-sided estimate; in fact, this is donein the submitted PhD thesis of the second author.We now have all ingredients at our disposal to derive the geometric observa-tion, formulated in the introduction for convex sets, in full generality.9 .11 Lemma.
Suppose A ⊂ R n , < q < ∞ , m is an integer, ≤ m < n , W isthe set of y ∈ dmn ξ A satisfying < dist( y, A ) < q and q | y − ξ A ( y ) | − ( y − ξ A ( y )) ∈ Dis( A, ξ A ( y )) ,x ∈ W , a = ξ A ( x ) , dim Dis( A, a ) ≥ n − m , U is an m dimensional subspaceof R n , U ⊂ Tan(
A, a ) , and q | x − a | − ( x − a ) belongs to the relative interior of Dis(
A, a ) . Then, lim sup W ∋ y → x | y − x | − dist( ξ A ( y ) − a, U ) < ∞ . Proof.
Assume a = 0, choose s and r such that 0 < s < | x | < r < q , and let Q = aff Dis( A, X of all v ∈ Q ∼{ } , such that q | v | − v belongsto the relative interior of Dis( A, Q and x ∈ X . Thisimplies the existence of ε > C = Q ∩ { v : | rv − x | < ε for some 0 < r < ∞} satisfies C ∩ { v : | v | = | x |} ⊂ X , hence qv ∈ Dis( A,
0) whenever v ∈ C ∩ S n − ;in particular, C ⊂ Nor( A,
0) by 4.2. We note that dim C = dim Q ≥ n − m andthat x belongs to the relative interior of C , as Q ∩ U ( x, ε ) ⊂ C . Abbreviating D = R n ∩ { d : d • c ≤ c ∈ C } , we observe U ⊂ D from [Fed59, 4.5], andemploying 0 ≤ γ < ∞ from 3.13 with v = x , we estimatedist( ξ A ( y ) , U ) ≤ − γ ξ A ( y ) • x + (1 + γ | x | ) dist( ξ A ( y ) , D ) ≤ γκ | x || y − x | + (1 + γ | x | )(2 q ) − | ξ A ( y ) | ≤ λ | y − x | whenever y ∈ W and s ≤ dist( y, A ) ≤ r by 3.14, 4.9, 4.7 (3), and 4.7 (2), where κ = (2 s ) − (1 + 2 q/ ( q − r )) and λ = γκ | x | + (1 + γ | x | )2 − q ( q − r ) − . Finally, x belongs to the interior of W ∩ { y : s ≤ dist( y, A ) ≤ r } relative to W by 3.2.Finally, we establish the structural theorem on the singularities of closedsets; in fact, we may formulate it for arbitrary subsets of Euclidean space. Suppose A ⊂ R n , m is an integer, and ≤ m ≤ n . Then, { a : dim Dis( A, a ) ≥ n − m } is a countably m rectifiable Borel set which can be H m almost covered by theunion of a countable family of m dimensional submanifolds of R n of class .Proof. Let B = { a : dim Dis( A, a ) ≥ n − m } . We assume A to be a nonemptyclosed set by 4.2, and also m < n . As 0 ∈ Dis(
A, a ) for a ∈ A by 4.2, we obtaindim Dis( A, a ) = dim(Dis(
A, a ) ∩ B (0 , a ∈ A from 3.9 (2); in particular, B is a Borel set by 3.10 (3) and 4.3. We define N tobe the set of all ( a, v ) ∈ A × R n such that v belongs to the relative interior ofDis( A, a ) ∩ B (0 , N is a Borel set by 3.11 and 4.3. By 4.4, we have ξ A ( x + v ) = a whenever ( a, v ) ∈ N . W i to be the Borel set of all x ∈ ξ − A [ B ] satisfying0 < dist( x, A ) < i − and (cid:0) ξ A ( x ) , i − | x − ξ A ( x ) | − ( x − ξ A ( x )) (cid:1) ∈ N for every positive integer i . Then, ξ A | W i is locally Lipschitzian by 4.7 (2) and3.2, and ( ξ A ( x ) , x − ξ A ( x )) ∈ N for x ∈ W i by 3.8. We observe that this implies that H n − m (cid:0) ( ξ A | W i ) − [ { ξ A ( x ) } ] (cid:1) > x ∈ W i , since ( ξ A | W i ) − [ { ξ A ( x ) } ] is relatively open in { ξ A ( x )+ v : v ∈ aff Dis( A, ξ A ( x )) } .We choose a countable family F of m dimensional affine planes in R n suchthat Q ∩ S F is dense in Q , whenever Q is an affine subspace of R n withdim Q ≥ n − m ; in fact, one may take F to be a countable dense subset in thefamily of all m dimensional affine planes in R n . Thence, we deduce, employing3.8, that B = S ∞ i =1 ξ A [ W i ∩ S F ];in fact, whenever a ∈ B , we take Q = { a + v : v ∈ aff Dis( A, a ) } , pick a positiveinteger i such that, for some x ∈ Q with 0 < | x − a | < i − , we have that i − | x − a | − ( x − a ) belongs to the relative interior of Dis( A, a ) ∩ B (0 , x within S F , and conclude x ∈ W i with ξ A ( x ) = a , as ( a, x − a ) ∈ N . Itfollows that B is countably m rectifiable.To prove the remaining property of B , we assume m ≥
1. Then, in viewof 2.7 and 4.11, we may apply 2.5 with f = ξ A | W i for every positive integer i to obtain the conclusion. Our proof of the countable m rectifiability follows [Fed59, 4.15 (3)],where the case of sets of positive reach was treated. If A is a closed convex set, this property was proven, by differentmethods, in [Alb94, Theorem 3]; the agreement, in this case, of the normalbundle used there with our distance bundle follows from 3.9 (1) (3) and 4.4. For 1 ≤ m < n , the preceding theorem may not be strengthenedby replacing the distance bundle by the normal bundle, as is evident from con-sidering a closed m dimensional submanifold of R n of class 1 that meets every m dimensional submanifold of R n of class 2 in a set of H m measure zero; theexistence of such A follows from [Koh77]. References [Alb94] Giovanni Alberti. On the structure of singular sets of convex func-tions.
Calc. Var. Partial Differential Equations , 2(1):17–27, 1994.URL: http://dx.doi.org/10.1007/BF01234313 .[Alm86] F. Almgren. Optimal isoperimetric inequalities.
In-diana Univ. Math. J. , 35(3):451–547, 1986. URL: http://dx.doi.org/10.1512/iumj.1986.35.35028 .[CV77] C. Castaing and M. Valadier.
Convex analysis and mea-surable multifunctions . Lecture Notes in Mathematics,Vol. 580. Springer-Verlag, Berlin-New York, 1977. URL: http://dx.doi.org/10.1007/BFb0087685 .11Fed59] Herbert Federer. Curvature measures.
Trans.Amer. Math. Soc. , 93:418–491, 1959. URL: https://doi.org/10.1090/S0002-9947-1959-0110078-1 .[Fed69] Herbert Federer.
Geometric measure theory . DieGrundlehren der mathematischen Wissenschaften, Band 153.Springer-Verlag New York Inc., New York, 1969. URL: http://dx.doi.org/10.1007/978-3-642-62010-2 .[HLW04] Daniel Hug, G¨unter Last, and Wolfgang Weil. A local Steiner-type for-mula for general closed sets and applications.
Math. Z. , 246(1-2):237–272, 2004. URL: http://dx.doi.org/10.1007/s00209-003-0597-9 .[Kel75] John L. Kelley.
General topology . Springer-Verlag, New York, 1975.Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], GraduateTexts in Mathematics, No. 27.[Koh77] Robert V. Kohn. An example concerning approximate differen-tiation.
Indiana Univ. Math. J. , 26(2):393–397, 1977. URL: .[KS80] David Kinderlehrer and Guido Stampacchia.
An introduction to vari-ational inequalities and their applications , volume 88 of
Pure and Ap-plied Mathematics . Academic Press Inc. [Harcourt Brace JovanovichPublishers], New York, 1980.[RZ01] J. Rataj and M. Z¨ahle. Curvatures and currents for unions of setswith positive reach. II.
Ann. Global Anal. Geom. , 20(1):1–21, 2001.URL: http://dx.doi.org/10.1023/A:1010624214933 .[San17] Mario Santilli. Rectifiability and approximate differentiability ofhigher order for sets, 2017. arXiv:1701.07286v1 .[Sch09] Reiner Sch¨atzle. Lower semicontinuity of the Willmore func-tional for currents.
J. Differential Geom. , 81(2):437–456, 2009. URL: http://projecteuclid.org/getRecord?id=euclid.jdg/1231856266 .[Sch14] Rolf Schneider.
Convex bodies: the Brunn-Minkowski theory , vol-ume 151 of
Encyclopedia of Mathematics and its Applications . Cam-bridge University Press, Cambridge, expanded edition, 2014. URL: https://doi.org/10.1017/CBO9781139003858 .[Sta79] L. L. Stach´o. On curvature measures.
Acta Sci. Math. (Szeged) , 41(1-2):191–207, 1979.[Z¨ah86] M. Z¨ahle. Integral and current representation of Federer’s curva-ture measures.
Arch. Math. (Basel) , 46(6):557–567, 1986. URL: http://dx.doi.org/10.1007/BF01195026 . Affiliations
Ulrich MenneInstitut f¨ur MathematikMathematisch-naturwissenschaftliche Fakult¨at12niversit¨at Z¨urichWinterthurerstrasse 190
Mario SantilliMax Planck Institute for Gravitational Physics (Albert Einstein Institute)Am M¨uhlenberg 114476
GolmGermany
University of PotsdamInstitute for MathematicsOT GolmKarl-Liebknecht-Straße 24–2514476