Linear bounds for constants in Gromov's systolic inequality and related results
aa r X i v : . [ m a t h . M G ] S e p Linear bounds for constants in Gromov’s systolic inequalityand related resultsAlexander NabutovskyAbstract.
Let M n be a closed Riemannian manifold. Larry Guth proved that there exists c ( n ) with the following property: if for some r > r is less than ( rc ( n ) ) n , then there exists a continuous map from M n to a ( n − r in M n . It was previously proven by Gromov thatthis result implies two famous Gromov’s inequalities: F illRad ( M n ) ≤ c ( n ) vol ( M n ) n and, if M n is essential, then also sys ( M n ) ≤ c ( n ) vol ( M n ) n with the same constant c ( n ). Here sys ( M n ) denotes the length of a shortest non-contractible closed curvein M n .Here we prove that these results hold with c ( n ) = ( n !2 ) n ≤ n . We demonstratethat for essential Riemannian manifolds sys ( M n ) ≤ n vol n ( M n ). All previouslyknown upper bounds for c ( n ) were exponential in n .Moreover, we present a qualitative improvement: In Guth’s theorem the assump-tion that the volume of every metric ball of radius r is less than ( rc ( n ) ) n can bereplaced by a weaker assumption that for every point x ∈ M n there exists a positive ̺ ( x ) ≤ r such that the volume of the metric ball of radius ̺ ( x ) centered at x is lessthan ( ̺ ( x ) c ( n ) ) n (for c ( n ) = ( n !2 ) n ).Also, if X is a boundedly compact metric space such that for some r > n ≥ n -dimensional Hausdorff content of each metric ball of radius r in X is less than ( r n +2) ) n , then there exists a continuous map from X to a ( n − r .1. Introduction.
Given a metric space its Kuratowski embedding into L ∞ ( X ) sends each point x to the distance function to x . In [Gr] Gromov defined the filling radius of a closedRiemannian manifold M n , F illRad ( M n ) as the infimum of r such that the imageof M n in L ∞ ( M n ) under the Kuratowski embedding bounds in its r -neighbourhood.In [Gr] Gromov gave a proof of the inequality F illRad ( M n ) ≤ c ( n ) vol ( M n ) n withthe constant that behaves as ( Cn ) n . (On the other hand Misha Katz’s paper [K] contains a short proof the inequality F illRad ( M n ) ≤ diam ( M n ) with the optimalconstant.) Gromov’s proof was later somewhat simplified by Stefan Wenger ([W]).An n -dimensional simplicial complex X n is essential , if there is no map f : X n −→ K ( π ( X n ) ,
1) such that f induces the isomorphism of the fundamental groups, andthe image of f is contained in the ( n − K ( π ( X n ) , X n is essential if the classifying map X n −→ K ( π ( X n ) ,
1) induces thehomomorphism of n th homology groups with non-trivial image (for some group ofcoefficients).The paper [Gr] contains a short and elegant proof of the inequality sys ( M n ) ≤ F illRad ( M n ) for all essential closed Riemannian manifolds (Lemma 1.2.B in [Gr]).One can define the ( n − Urysohn width , U W n − ( X ), of a metricspace X as infimum of t such that there exists a continuous map f : X −→ K n − to a( n − K n − such that for each k ∈ K n − f − ( k ) has diam-eter ≤ t . We will need also a closely related notion of Alexandrov width , U R n − ( X ),that is defined almost as the Urysohn width, but with condition diam ( f − ( k )) ≤ t replaced by the condition that f − ( x ) is contained in a metric ball of radius t . Itis obvious that U W n − ( X ) ≤ U R n − ( X ). Alexandrov width of a compact metricspace X can also be defined as the infimum of t such that there exists a covering of X by connected open sets U α of diameter ≤ t such that no ( m + 1) sets U α havea non-empty intersection. To see that these two definitions are equivalent for allcompact X , denote U R n − ( X ) in the sense of the first definition by ̺ , and the sec-ond by r . We will first demonstrate that r ≤ ̺ + ε for an arbitrarily small positive ε , and then demonstrate that ̺ ≤ r . To prove the first inequality choose K n − and f such that max k ∈ K n − diam ( f − ( k )) is very close to ̺ . Now consider a veryfine covering of K n − by open sets V β such that each ( n + 1)-tuple of these setshas the empty intersection. Finally, define the collection U α as the collection of allconnected components of open sets f − ( V β ). Note that the diameters of U α do notexceed max k ∈ K n − diam ( f − ( k )) + ε , where ε > V β sufficiently fine. To see that ̺ ≤ r , consider the nerve K n − of the covering U α , and the standard map f : X −→ K n − defined using a partitionof unity subordinate to the covering { U α } . Now note that the inverse images of eachpoint of K n − under f will be contained in one of the sets U α .Gromov also provided a proof of the inequalities F illRad ( M n ) ≤ U W n − ( M n )(the combination of Proposition (D) in Appendix 1 in [Gr] with the inequality in theexample at the end of section (B) in Appendix 1 of [Gr]). Therefore, any upper boundfor U W n − automatically leads to upper bounds to F illRad and, in the essentialcase, for sys . Now a natural question (posed by Gromov in [Gr]) is whether ornot U W n − ( M n ) ≤ c ( n ) vol ( M n ) n . This question was solved in the affirmative byLarry Guth in [Gu11], [Gu17]. In fact, Guth proved more. He demonstrated that there exists δ ( n ) such that if for some r > r have volumeless than δ ( n ) r n , then U W n − ( M n ) ≤ r . To recover the previous inequality one cantake here r = vol ( M n ) n δ ( n ) n . The assumption will automatically hold, and one sees that U W n − ( M n ) ≤ δ ( n ) − n vol ( M n ) n .Recall that the m -dimensional Hausdorff content of a compact metric space X isthe infimum over all coverings of X by metric balls with radii r i of the sum P i r mi . (Ifone requires here that all r i do not exceed δ , and then takes the limit as δ −→
0, oneobtains the m -dimensional Hausdorff measure of X .) Guth asked if one can replacethe volume in these inequalities by the n -dimensional Hausdorff content, and if suchestimates will be true for all (not necessarily n -dimensional) compact metric spaces(Questions 5.1 and 5.2 in [Gu 17]).In [LLNR] we proved that this is, indeed, so. For example, we proved that for eachcompact metric space X and each integer m > U W m − ( X ) ≤ C ( m ) HC m m ( Y ). Asa corollary, we immediately see that if X is a compact m -essential smooth polyhe-dron endowed with the structure of the length space, sys ( X ) ≤ C ( m ) HC m ( X ).Recently, Panos Papasoglu wrote a paper [P] with a much shorter proof of theseresults than the proof in [LLNR]. His proof did not contain an estimate for C ( m ),but our analysis of his proof yields C ( m ) ∼ const m leading to exponential in m estimates for constants in the previous inequalities. He learned about [LLNR] frommy talk and our conversations at the conference at Barcelona. While his proof drawson several ideas of [LLNR], it also contains a central observation that is quite dif-ferent from the ideas of [LLNR]. Roughly speaking, the amazing in its strength andsimplicity Papasoglu’s insight was to consider an (almost) minimal “hypersurface”dividing a compact metric space into subsets of a small diameter and to observe thatthe “area” (or, more precisely, the appropriate Hausdorff content) of the intersectionof this hypersurface with any metric ball cannot exceed the Hausdorff content ofthe corresponding metric sphere. Indeed, otherwise, one could just replace the partof the minimal hypersurface inside the metric ball by the metric sphere preservingthe same upper bound for diameter for each component of the complement. Thus,the (almost) minimal hypersurface inherits the main property of the metric space,namely, that its intersections with metric balls of a certain size are “small”. Thisobservation enables one to run an induction argument, where the result for the met-ric space and the n -dimensional Hausdorff content would follow from the same resultfor the minimal hypersurface and its ( n − sys ( M n ) ≤ nvol ( M n ) n for Riemannian tori. The proof of our main theorem below is heavily based on Papasoglu’s idea. Yet it contains a number ofmodifications and simplifications:First, we observed that the dependence of the constant in the inequalities couldbe improved from exponential to linear by (a) carefully choosing the radius of themetric ball (in the argument above) and (b) improving an argument of the end of theproof of Lemma 2.4 so as not to decrease the constant by a constant factor on eachstep - compare our Lemma 2.5, where the upper bound in the assumption and theconclusion is the same. (We observed that it is more convenient to use U R insteadof the previously used
U W here.) In order to accomplish (a), one could use the trickused by Larry Guth at the end of [Gu10] (as it was done in the first version of thepresent paper).However, we noticed that there is a better (point dependent) way to choose theradii as in Lemma 2.4 below. Not only this observation leads to an improvement ofthe estimate by a constant factor, but it also yields a quantitative improvement ofall the previous results that was mentioned in the abstract: the radius of a smallball centered at a point is allowed to depend on the center as long as it does notexceed a fixed r . It is interesting to note that we do not see how to achieve thisquantitative improvement, if one follows the approach of [P] via Hausdorff contents,as in this approach needs to restrict the radii of the considered balls by a quantitythat depends on the radii of small balls r and becomes wildly variable, if these radiiare allowed to depend on the centers. (See the remark at the end of section 3 formore details.)Second, the approach of Papasoglu to the classical systolic geometry was throughresults about Hausdorff contents (the same as in [LLNR]). He noticed that instead onecan directly use the Hausdorff measure and the Eilenberg’s inequality. We adoptedthis approach and discovered that not only it leads to a much simpler proof, but onecan save a √ n factor as the coarea inequality for Riemannian polyhedra is strongerthan Eilenberg’s inequality.Third, we were careful about the values of the constants. This is probably notthat important in the long run, as one expects that the dimensional constants inthe above inequalities should behave as const √ n and not as const n . Still, as theresult, we derive aesthetically pleasing and convenient to use inequalities sys ( M n ) ≤ n vol n ( M n ) for all closed essential manifolds, and F illRad ( M n ) ≤ n vol n ( M n ) forall closed manifolds. In fact, I am not aware of any previously published specificvalue of the constant at vol ( M n ) n in the general case of Gromov’s systolic inequalityfor n = 3 other than Gromov’s 1296 √ n = 3 our value2 ∗ = 2 . . . . of the systolic constant is within the factor of 2 of the (unknown)optimal value. Fifth, in the last section of the paper we also similarly improve the main result of[LLNR] and [P]. We provide a linear in m bound for c ( m ) with a reasonable constantin the inequality U W m − ( X ) ≤ c ( m ) HC m m ( X ) for a compact metric space X as wellas a related bound for the constant in the local version of this result for boundedlycompact X . This part of our paper is much closer to [P].Our first main theorem is: Theorem 1.1.
Let M n be a closed Riemannian manifold, and r > a real number.Assume that for every x ∈ M n there exists t = t ( x ) ∈ (0 , r ] such that the volume ofthe metric ball of radius t centered at x is less than t n n ! . Then U R n − ( M n ) < r. The relationships between
U R n − , U W n − , F illRad and sys stated above imme-diately imply that: Theorem 1.2.
Assume that M n is a compact n -dimensional smooth Riemannianmanifold. Then U R n − ( M n ) ≤ ( n !2 ) n vol ( M n ) n ≤ n vol n ( M n ) , (1) U W n − ( M n ) ≤ n !2 ) n vol ( M n ) n ≤ n vol n ( M n ) , (2) F illRad ( M n ) ≤ ( n !2 ) n vol ( M n ) n ≤ n vol n ( M n ) , (3)Inequality (1) and the inequality sys ( X n ) ≤ U R n − X n ) for essential Riemannianpolyhedra which is a combination of two inequalities: sys ( X n ) ≤ F illRad ( X n )and F illRad ( X n ) ≤ U R n − ( X n ) that were proven in [Gr] immediately imply the fol-lowing version of Gromov’s systolic inequality with linear in n dimensional constant:For each essential Riemannian manifold M n sys ( M n ) ≤ nvol n ( M n ) that appearedin the first version of the present paper. However, Roman Karasev e-mailed to me avery short proof of a stronger inequality: sys ( X n ) ≤ U R n − ( X n ) . (4)for essential polyhedral length spaces that does not involve the Kuratowski embed-ding or the filling radius. Karasev’s proof is based on work of Albert Schwartz [S],and in a nutshell goes as follows: Theorem 14 in [S] implies that if M n is essen-tial, then there is no open cover of M n by connected open sets U α such that: 1)multiplicity of intersections of U α does not exceed n ; 2) If p : ˜ M n −→ M n is theuniversal covering of M n , then the restriction of p to each connected component of p − ( U α ) is a homeomorphism. On the other hand the second definition of Alexandrovwidth implies that there is an open covering of M n by connected open sets of radius ≤ U R n − ( M n ) with multiplicity of intersections ≤ n . If sys ( M n ) > U R n − ( M n ),then each loop in U α is contractible in M n . Therefore, { U α } is an open covering of ˜ M n the existence of which contradicts the conclusion of Theorem 14 of [S]. WithKarasev’s permission I will present a self-contained proof of inequality (4) at the endof this paper.As an immediate corollary: Theorem 1.3. If M n is a closed essential Riemannian manifold, then sys ( M n ) ≤ n !2 ) n vol n ( M n ) ≤ n vol n ( M n ) . Remark.
As ( n !) n = e (1 + o (1)) n, for all sufficiently large n sys ( M n ) < . n vol ( M n ) n . If n = 2, the inequality in the theorem is well-known, and abetter estimate can be found in section 1.4.3 of [BZ]. If n = 3, then the constant at vol ( M ) in Theorem 1.3 is equal to 2 ∗ = 2 . . . . . On the other hand, we seethat the optimal value of this constant for n = 3 cannot be less than π = 1 . . . . ,as this is the value that one gets in the case of RP with the canonical metric. So, for n = 3, our constant is within the factor of 1 .
97 from the optimal systolic constant.We will prove Theorems 1.1, 1.2 in a somewhat greater generality, namely for com-pact Riemannian polyhedra (i.e. finite polyhedra endowed with smooth Riemannianmetric on each maximal simplex, so that Riemannian metric on two simplices thathave a common face match on this face). Note that all previous definitions andquoted results by Gromov can be directly extended to Riemannian polyhedra, whichwas observed by Gromov in [Gr].Below a subpolyhedron will always mean a compact subpolyhedron with smoothlyembedded faces endowed with the the Riemannian metric of the ambient Riemannianpolyhedron (and the corresponding intrinsic distance). Also, below | X | will denotethe volume of X . Sometimes we write it as | X | n , when we want to emphasize thedimension.Recall that a metric space is boundedly compact, if every closed subset of X iscompact. In the last section, we prove that: Theorem 1.4.
1. Let X be a compact metric space, r > , n a positive inte-ger. Assume that for each metric ball B of radius r HC n ( B ) < ( r n +2) ) n . Then U R n − ( X ) ≤ r .2. Let X be a compact metric space, n ≥ . Then U R n − ( X ) ≤ n +2) HC n ( X ) n ≤ nHC n ( X ) n .3. Let X be boundedly compact. Assume that for each metric ball B of radius rHC n ( B ) < ( r n +2) ) n . Then U R n − ( X ) ≤ r . Remark. As U W n − ( X ) ≤ U R n − ( X ), we also immediately obtain the correspond-ing upper bounds for the Urysohn width of X , U W n − ( X ), that differ from the upperbounds for U R n − ( X ) by a factor of 2. Proof of Theorems 1.1 and 1.3.
The well-known coarea inequality for Lipschitz functions of Riemannian manifoldsimmediately generalizes to Riemannian polyhedra ([BZ]) and implies that given aRiemannian polyhedron X n , a real r , and a metric ball B of radius r centered at apoint x ∈ X n , R r | S s | n − ds ≤ | B | n , where S s denotes the metric sphere of radius s centered at x .We prefer to work in the situation when for almost all s S s is a subpolyhedron. Onewell-known way to achieve this is to approximate the distance function by (1 + τ )-Lipschitz function that is smooth on each open simplex (cf. section 3 of [Ga]), andto replace the distance function by this approximation. (Here τ can be chosen to bearbitrarily small.) Starting from Lemma 2.2 “metric spheres” will really mean thelevel sets of a sufficiently close smooth approximation of the distance function. NowSard’s theorem implies that almost all geodesic spheres are subpolyhedra. (Sard’stheorem will separately apply to the restriction of the smooth approximation of thedistance function to each open simplex.) However, the coarea inequality above andall inequalities below will hold only up to a factor of 1 + f ( τ ), where f will be somespecific function such that lim τ −→ f ( τ ) = 0. Eventually, one will pass to the limitas τ −→
0. For the sake of readability we will not be mentioning terms of the form1+ f ( τ ) in the inequalities, and will just pretend that the distance function is smoothon each open simplex. Lemma 2.1.
Let X be a compact Riemannian polyhedron of dimension ≤ . Assumethat there exists r > such that for each x ∈ X there exists a metric ball B centeredat x of radius t ( x ) ∈ (0 , r ] such that | B | < t ( x ) . Then U R ( X ) < r . In other wordseach connected component of X can be covered by a a metric ball of radius < r .Proof. First, note that without any loss of generality we can assume that X is con-nected.Second, observe that the lemma can be reduced to its particular case, when X is tree (endowed with Riemannian metric on each edge). Indeed, by disconnectingsome of the edges of X from one of their endpoints to destroy cycles in X we cantransform X into a Riemannian tree Y . We also have a (quotient) map f : Y −→ X ,obtained by identifying new vertices of Y with the corresponding old ones. Thedistances in Y are not less than distances between the images of the same points in X . Therefore, each metric ball with center y in Y is a subset of the metric ball in X with the center f ( y ) and the same radius. Therefore, the assumption of the lemmaholds for Y . On the other, if Y can be covered by a metric ball with center y , thenthe metric ball in X with the center f ( x ) and the same radius will cover X .Therefore, we can assume that X is a Riemannian tree. Let p be a center of X ,that is, a point in Y realizing the minimum, R , of max q ∈ X dist X ( p, q ). Let x ∈ Y denote one of the most distant points of Y from p (in the metric of Y ). (So, R isthe radius of X .) As X is contained in the metric ball of radius R centered at p , weneed to prove that R < r .Observe that the definition of p implies that there exists another point x ′ ∈ Y suchthat dist Y ( p, x ′ ) = dist Y ( p, x ), and the (unique) shortest path from x to x ′ passesthrough p (as Y is a tree). Therefore, dist ( x, x ′ ) = 2 dist ( p, x ) = 2 R . This impliesthat for each t ≤ R the length of X ∩ B ( x, t ) ≥ t , and therefore R < r . (cid:3) Lemma 2.2.
Assume that B is a metric ball of radius r centered at x ∈ X n +1 , ε ∈ (0 , . Assume that | B | ≤ cr n +1 (1 − ε ) for some c . Let λ = r ( ε ) n +1 . There existsa subset A of open interval ( λ, r ) of positive measure such that for each metric sphere S centered at x with radius t ∈ A , | S | < ( n + 1) ct n (1 − ε ) , and S is a subpolyhedronof X n +1 .Proof. Assume that the set of radii > λ such that | S | < ( n +1) ct n (1 − ε ) has measurezero. The coarea inequality implies | B | ≥ R rλ c ( n + 1) t n (1 − ε/ dt > cr n +1 (1 − ε/ − cλ n +1 = cr n +1 (1 − ε ), yielding a contradiction with our assumption. To ensurethat S is a subpolyhedron, we apply Sard’s theorem on each open simplex of X n +1 .(Recall, that by distance function we actually mean a smooth approximation to thedistance function.) (cid:3) Definition 2.3.
A compact subpolyhedron Y m − of X m is called d -separating if eachconnected component of its complement X m \ Y m − can be covered by a metric ballof radius ≤ d . Denote the infimum of | Y | m − over all d -separating sets Y in X m by I X ( d, m − δ > d -separating set Y m − is called δ -minimal if | Y m − | ≤ I X ( d, m −
1) + δ . Lemma 2.4.
Assume that X n +1 is a Riemannian polyhedron of dimension ≤ n + 1 such that for some positive r , ε and τ each each x ∈ X n +1 there exists t ( x ) ∈ ( τ, r ] such that the metric ball B of radius t ( x ) centered at x satisfies the inequality | B | n +1 < t ( x ) n +1 ( n +1)! (1 − ε ) . Then there exists δ = δ ( n, ε, τ ) such that for every δ -minimal r -separating set Z and each x ∈ X n +1 there exists ̺ ∈ ( t ( x )( ε ) n +1 , t ( x )) such that:(a) The metric ball β in X n +1 of radius ̺ centered at x satisfies | Z \ β | n < ̺ n n ! (1 − ε (b) If x ∈ Z and α denote the metric ball in Z of radius ̺ centered at x , where Z isendowed with the intrinsic metric, then the volume of α is less than ̺ n n ! (1 − ε ) .Proof. The distances between two points of Z endowed with the inner metric cannotbe less than the distance between these points in the metric of X n +1 . Therefore, assertion (b) of the lemma follows from assertion (a) simply because each metric ballin Z endowed with the inner metric is contained in the metric ball in X n +1 with thesame center and the same radius. Thus, it is sufficient to prove (a).We apply the previous lemma to B . There exists a metric sphere S of radius s ∈ ( t ( x )( ε ) n +1 , t ( x )) that is a subpolyhedron and satisfies | S | < (1 − ε ) s n n ! . Let λ = τ ( ε ) n +1 , δ = ελ n n ! . Observe that2 s n n ! (1 − ε ) − | S | > s n n ! ε > δ. ( ∗ ) . The proof is by contradiction. Let Z be a δ -minimal r -separating set. We assumethat for some x and each ̺ ∈ ( t ( x )( ε ) n +1 , t ( x )) the metric ball β or radius ̺ centeredat x does not satisfy the inequality in part (a) of the lemma. In particular, this istrue for the metric ball β of radius s bounded by S : | Z T β | ≥ s n n ! (1 − ε ), We aregoing to modify Z to obtain another r -separating set Z ′ with volume less than | Z | − δ arriving to a contradiction.To construct Z ′ we remove from Z all points inside the metric ball bounded by S ,and take the union of the resulting set Z with S . It is obvious that Z ′ is r -separating.Indeed, all components of X n +1 \ Z outside of S became smaller or unchanged whenwe replace Z by Z ′ , and the “new” component or components of X n +1 \ Z ′ inside of S can clearly be covered by the metric ball of radius r centered at x .It follows from formula (*) that | Z ′ | < | Z | − δ . (cid:3) Lemma 2.5.
Assume that Y n is a d -separating subpolyhedron in X n +1 such that forsome d U R n − ( Y n ) ≤ d . Then U R n ( X n +1 ) ≤ d .Proof. First proof. Let { U α } α ∈ A be the set of all connected components of X \ Y .Observe that for each α ∂U α ⊂ Y . Consider the map f of Y to a ( n − K such that for each x ∈ K f − ( x ) can be covered by a metricball b ( x ) of radius d + δ in the intrinsic metric of Y n , where δ is arbitrarily small.Observe that the metric ball in X n +1 with the same center and radius will contain b ( x ), and, therefore, f − ( x ). For each α take a copy CK α of the cone CK over K .Using the version of Tietze extension theorem for maps into contractible simplicialcomplexes, we can extend the restriction of f to ∂U α to a continuous map g of theclosure of U α to CK α (compare with a similar argument in section 6.1 of [LLNR]).We would like to change this map so that the images of all points of of ∂U wouldremain unchanged, and all points in ¯ U α \ ∂U α will be mapped to ( CK ) α \ K α (thatis, to the interior of the cone). For this purpose endow each top-dimensional simplexin CK α by the metric of the Euclidean regular simplex with side length d . For each x ∈ ¯ U α g ( x ) will be either the tip of the cone, or a point on the unique generatorof the cone passing through x (which is a straight line segment with one end at K α and another end at the tip of the cone). If g ( x ) is the tip of the cone, the new map h ( x ) = g ( x ). Otherwise, let φ ( x ) denote the distance from g ( x ) to the tip of thecone. Now move g ( x ) towards the tip of the cone by min { φ ( x ) , dist ( x, ∂U ) } . Nowglue all copies of CK α into one n -dimensional simplicial complex L by identifying allcopies of K α at the boundaries into one copy of K .The resulting map h : X −→ L is a continuous map. By construction, its re-striction to Y coincides with f , and for each point x ∈ Y f − ( f ( x )) is in Y . Foreach x ∈ U α f ( x ) ∈ CK α \ K , and f − ( f ( x )) ∈ U α . In both cases f − ( f ( x )) can becovered by a ball of radius d + δ . Second proof.
After reading the first version of this paper ([N]) Roman Karasevsuggested the following very simple proof of Lemma 2.5. His proof uses the definitionof
U R n ( X ) is the infimum of r such that there exists a cover of X by open sets withradii ≤ r with multiplicity of the covering ≤ n + 1. Start with an open covering of Y n of multiplicity ≤ n with radii of the sets in the intrinsic metric ≤ d . Then we convertthis covering of Y n into an open covering of a very small open neighbourhood of Y n in X n +1 without increasing the multiplicity and increasing the radii by not morethan an arbitrarily small amount. Finally, we add all open sets U α to the coveringincreasing its multiplicity by 1. (cid:3) Proposition 2.6.
Let r, ε, τ < r be positive real numbers, and X n is a compact n -dimensional Riemannian polyhedron. Assume that for each x ∈ X n there exists t ( x ) ∈ ( τ, r ] such that the metric ball B in X n of radius t ( x ) centered at x satisfies | B | < t ( x ) n n ! (1 − ε ) .Then U R n − ( X ) < r .Proof. We are going to prove this proposition using the induction with respect tothe dimension n . Lemma 2.1 is the base of induction. To prove the induction stepassume that the theorem is true for n . To prove it for n + 1 choose a sufficiently smallpositive δ (as in Lemma 2.4) and consider a δ -minimal r -separating subpolyhedron Z . Lemma 2.4 implies that Riemannian subpolyhedron Z satisfies the assumptionsof the theorem for n , ̺ ∈ ( τ ( ε ) n +1 , r ] as s new value of t ( x ), ε as a new value of ε , and τ ( ε ) n +1 as a new value of τ . The induction assumption implies that U R n − ( Z ) ≤ ̺ < r in the intrinsic metric on Z . The same inequality will automatically be truefor the “shorter” extrinsic metric. Now the induction step follows from Lemma 2.5applied for Y = Z , d = r . (cid:3) Now we are going to establish Theorem 1.1 for the class of all compact Riemannianpolyhedra (and not only Riemannian manifolds):
Theorem 2.7.
Let M n be a compact Riemannian polyhedron (for example, a closedRiemannian manifold), and r > a real number. Assume that for every x ∈ M n there exists t = t ( x ) ∈ (0 , r ] such that the volume of the metric ball of radius t centered at x is less than t n n ! . Then U R n − ( M n ) < r. Proof.
We are going to deduce this theorem from Proposition 2.6 by proving that itsassumption can be relaxed in two ways.First, we would like to demonstrate that the assumption of existence of ε > x there exists t ∈ [ τ, r ] such that | B | < t ( x ) n n ! (1 − ε ) is equivalent tothe assumption that for each x there exists t ∈ [ τ, r ] such that | B | < t ( x ) n n ! . Observethat | B ( x,R ) | R n is a continuous function of the center x ∈ M n of the ball B ( x, R ) and itsradius R . Note that ̺ ( x ) = min R ∈ [ τ,r ] | B ( x,R ) | R n is a continuous function of x such thatits value at every point is strictly less than n ! . Hence, the maximum of ̺ ( x ) over all x ∈ M n will be attained at some point, and will be strictly less than n ! . Now onecan choose ε as 0 . n ! − max x ∈ M n ̺ ( x )).Second, we are going to demonstrate that one does not need the condition thatthere exists τ > x and some t as in Proposition 2.6 t ≥ τ ,as this condition automatically holds. For all x ∈ M n formally define t ( x ) assup t ∈ (0 ,r ] { t | | B ( x,t ) | t n < n ! } . The assumption of the theorem is that the set of t isnon-empty, and, therefore, t ( x ) is defined and positive for all x . Observe that t ( x )is lower-semicontinuous, i.e. t ( x ) ≤ lim inf y −→ x t ( y ). Indeed, as | B ( x,t ) | t n is continuous,if | B ( x,t ) | t n < n ! for some t , then the same inequality will be true for all y sufficientlyclose to x . Therefore, for each positive δ the inequality t ( y ) ≥ t ( x ) − δ holds for all y sufficiently close to x . This observation immediately implies the lower-semicontinuityof t . Hence, t ( x ) attains its positive minimum on M n which can be chosen as τ . Nowthe theorem follows from Proposition 2.6. (cid:3) Proof of the inequality (4) : Finally, I am going to present an elementary proof ofthe inequality sys ( X n ) ≤ U R n − ( X n ) for essential polyhedral length spaces X n .I learned the idea of this proof from Roman Karasev. This proof is based on ideasfrom [S].Recall that U R n − ( X n ) can be defined as the lower bound of r such that thereexists a covering of X n by connected open sets with radii ≤ r with multiplicity ≤ n .Assume that sys ( X n ) > r , where r = U R n − ( X n ). Choose a covering of X n of multiplicity ≤ n by connected open sets U α with radii ≤ r + δ , where δ < / sys ( M n ) − r ). Consider a closed curve γ in some U α . Let p be thecenter of a ball of radius r + δ covering U α . We can homotope γ into a concatenationof many thin triangles pγ ( t i ) γ ( t i +1 ), where the length of the arc of γ between γ ( t i )and γ ( t i +1 ) does not exceed δ , and two other sides are minimizing geodesics. Thelength of each of these triangles is less than sys ( X n ). Therefore, these triangles arecontractible, and so is γ . Thus, the inclusion homomorphisms π ( U α ) −→ π ( X n )are trivial, and each U α lifts to a collection of disjoint open sets ( e U α ) g ⊂ ˜ X n , where g runs over π ( X n ), and e X n denotes the universal covering of X n endowed with thepullback metric.Consider the nerves N of the covering { U α } of X n , and e N of the covering { ( e U α ) g } of e X n . It is easy to see that there exists a commutative square with horizontal sides e X n −→ e N and X n −→ N , where vertical sides e X n −→ X n , and e N −→ N arethe universal covering maps. This easily implies that the map X n −→ N inducesan injective homomorphism π ( X n ) −→ π ( N ). As this homomorphism is alwayssurjective, it is an isomorphism. Thus, the classifying map X n −→ K ( π ( M n ) , N that has dimension ≤ n −
1. (Recall that all maps X n −→ K ( π ( X n ) ,
1) that induce the isomorphism of fundamental groups are homotopic.)Therefore, X n is not essential. Equivalently, if X n is essential, then sys ( X n ) ≤ r .3. Hausdorff content
Similar ideas can be applied to majorize
U R m − of a compact or even a boundedlycompact metric space X in terms of the Hausdorff content, HC m , of metric balls in X . The key is the coarea inequality proven in [LLNR].For convenience of the reader we reproduce the inequality and its (very shortproof) below: Lemma 3.1. ([LLNR]) Let m > , X is a compact metric space, B ⊂ X a metricball of radius r , S s , s ∈ (0 , r ] , metric spheres of radius s with the same center as B .There exists s ∈ (0 , r ) such that HC m − ( S s ) ≤ r HC m ( B ) .Proof. For an arbitrarily small ε > B by closed metricballs B i with radius r i such that P i r mi ≤ HC m ( X ) + ε . Observe that HC m − ( S s ) ≤ HC ∗ m − ( S s ) defined as P i ∈ I s r m − i , where I s is the set of indices i such that B i ∩ S s = ∅ .We are going to prove that for some s HC ∗ m − ( S s ) ≤ r P i r mi , which implies thelemma. For this purpose it is sufficient to prove that R r HC ∗ m − ( S s ) ds ≤ P i r mi . For each i and s define χ i ( s ) as 1, if B i ∩ S s = ∅ , and 0 otherwise. Using this nota-tion R r HC ∗ m − ( S s ) ds = R r P i r m − i χ i ( s ) ds = P i r m − i R r χ i ( s ) ds = P i r m − i (2 r i ) =2 P i r mi . (cid:3) Lemma 3.2.
Assume that for every metric ball B of radius r in X HC ( B ) < r .Then U R ( X ) < r .Proof. We are going to prove that X is a union of disjoint metric balls of radius ≤ r .If so, we can map X into the set of centers of this balls by sending each of thesemetric balls to its center.Given x ∈ X apply the coarea inequality (Lemma 3.1) to the ball of radius r centered at x . We are going to obtain a geodesic sphere S centered at x of radius < r such that HC ( S ) ≤ r HC ( B ( x, r )) < . Note that HC ( S ) is just the minimal number of metric balls in X required to cover S , and so it is either equal to 1, if X is non-empty, or to 0, if X is empty. Therefore, we conclude that HC ( S ) = 0, and S is empty. As X is closed, x is contained in a closed metric ball of radius strictlyless than the radius of S which is less than r . (cid:3) Definition 3.3.
Given a subset A of X , positive real n , and δ >
0, a collection ofmetric balls B i with radii r i is called a ( n, δ )-optimal covering of W , if they cover W , and P i r ni ≤ HC n ( W ) + δ. Define ε = , and for each n ≥ ε n +1 = ε n n +3) exp( − < ε n n +3) (1 − n +3 ) n .Obviously, ε n +1 = 2(2 exp(2)) − n n +3)! , and it is not difficult to see that n +3) <ε n +1 n +1 < n +3) . Lemma 3.4.
Assume that B is metric ball of radius r centered at x . There exists asphere S centered at x with radius ̺ ∈ [ r (1 − n +3 ) , r (1 − n +3 )] such that HC n ( S ) ≤ n + 3) HC n +1 ( B ) r .Proof. The lemma immediately follows from the coarea inequality (Lemma 3.1). (cid:3)
Definition 3.5.
A closed subset set Y of X is called d -separating if the each con-nected component of its complement X \ Y can be covered by a metric ball of radius ≤ d . Let HC ( b ) n ( Y ) denote the infimum of P i r ni over all coverings of Y by closedmetric balls with radii r i such that all radii r i do not exceed b . Denote the infimumof HC ( b ) n ( Y ) over all d -separating sets Y in X by I X ( d, b, n ). If δ > d -separating set Y is called ( b, n, δ )-minimal if HC ( b ) n ( Y ) ≤ I X ( d, b, n ) + δ .Using HC ( b ) n ( Y ) instead of HC n ( Y ) here is another simple but beautiful idea ofPapasoglu from [P] designed to overcome non-additivity of Hausdorff content (andstrongly reminiscent of ideas used to the same purpose in [LLNR]). Lemma 3.6.
Assume that for each metric ball B of radius r in X HC n +1 ( B ) ≤ ε n +1 r n +1 . Assume that Z is a ( r n +3) , n, δ ) -minimal r (1 − n +3) ) -separating set for asufficiently small positive δ = δ ( n, r ) . Then for each ball β of radius ̺ = r (1 − n +3 ) in X HC n ( Z \ β ) ≤ ε n ̺ n . Proof.
The proof is by contradiction. We assume that there exists a ball β or radius ̺ centered at a point x that does not satisfy the above inequality. We are going tomodify Z to obtain another r -separating set Z ′ with considerably lower HC ( r n +3) ) n than Z obtaining a contradiction proving the lemma. We start from choosing a sphere S centered at x as in the previous lemma. So, HC n ( S ) ≤ n + 3) ε n +1 r n . As 2( n + 3) ε n +1 ≤ n +3)) n , HC ( r n +3) ) n ( S ) = HC n ( S ).Also, the radius of S does not exceed r (1 − n +3 ) and is not less than r (1 − n +3 ).To construct Z ′ we remove from Z all points inside the ball bounded by S , and takethe union of the resulting set Z with S . It is obvious that Z ′ is r (1 − n +3 )-separating.Now we are going to estimate HC ( r n +3) ) n ( Z ). First, note that none of the ballsof radius ≤ r n +3) used to cover Z in a nearly optimal way can intersect the ball B ′ of radius r (1 − . n +3 ) centered at x . On the other hand, every metric ball of radius ≤ r n +3) that has a non-empty intersection with β is contained inside B ′ . Therefore, HC ( r n +1) ) n ( Z ) ≥ HC ( r n +1) ) n ( Z ) + HC ( r n +1) ) n ( Z ∩ β ) ≥ HC ( r n +1) ) n ( Z ) + HC n ( Z ∩ β ).Therefore, HC ( r n +3) ) n ( Z ) ≤ HC ( r n +3) ) n ( Z ) − HC n ( Z ∩ β ), and HC ( r n +3) ) n ( Z ′ ) ≤ HC ( r n +3) ) n ( Z ) − HC n ( Z ∩ β ) + HC n ( S ) ≤ HC ( r n +3) ) n ( Z ) − r n [ ε n (1 − n +3 ) n − n +3) ε n +1 ] < HC ( r n +3) ) n ( Z ) − δ for a sufficiently small positive δ = δ ( n, r ) . (cid:3) Lemma 3.7.
Assume that Y is a d -separating set in X such that for some dU R n − ( Y ) ≤ d . Then U R n ( X ) ≤ d .Proof. The only difference of this lemma from Lemma 2.5 is that we do not assumethat X is an n -dimensional Riemannian polyhedron, and Y its subpolyhedron ofcodimension 1. Yet either of the two proofs of Lemma 2.5 above can be used withoutany changes to prove the present lemma. (cid:3) Theorem 3.8.
Let r be a positive number, n a positive integer, X a compact metricspace such that for each metric ball B of radius r HC n ( B ) ≤ ε n r n , where ε n weredefined at the beginning of this section. Then U R n − ( X ) ≤ r. Proof.
We are going to use the induction with respect n . Lemma 3.2 provides thebase of induction. To prove the induction step assume that the theorem is true for n . To prove it for n + 1 choose a sufficiently small positive δ (as in Lemma 3.6)and consider a ( n +3) r, n + 1 , δ )-minimal r (1 − n +3 )-separating set in X . Lemma 3.6implies that Z satisfies the conditions of the present theorem for n and ̺ ≤ r (1 − n +3 ).The induction assumption implies that U R n − ( Z ) ≤ ̺ < r . Now the induction stepfollows from the previous lemma applied for Y = Z , d = r . (cid:3) Proof of Theorem 1.4. As ε n ≥ ( n +2) ) n , the first part of Theorem 1.4 follows. Toprove the second part it is sufficient to take r = ( HC n ( X ) ε n ) n and apply the first part.To prove the third part one uses another trick from [P]. Let ̺ > x of a boundedly compact X and covers X by two overlapping sets of closed annuli centered at x . One family of annuli involves radii in the intervals[8( i − ̺, i̺ ] for all positive integer i , another [(8( i −
1) + 4) ̺, (8 i + 4) ̺ ]. The ideais that the union of almost minimal ̺ -separating subsets for all these annuli willbe a ̺ -separating family for X . One can even remove the parts of almost minimalseparating sets in all annuli that bound a domain only together with a non-emptysubset of the boundary of the annulus. The remaining parts of almost minimalseparating sets in annuli satisfy the assumptions and conclusion of Lemma 3.6. Asthe result Lemma 3.6 will be almost true for the union of all these separating sets:We loose just a factor of 2 in the conclusion of Lemma 3.6 as we need to combinethe union of two families of disjoint separating subsets. This leads to appearance ofthe extra factor of 2 in the denominator of the right-hand side of the inequality inTheorem 1.4, part 3. Remark.
We do not see how to adapt this proof of Theorem 1.4(1) to prove itsversion, where the assumption that each metric ball B of radius r satisfies the in-equality HC n ( B ) ≤ n +20) n r n is replaced by a weaker assumption that for each r there exists ̺ ∈ (0 , r ) such that HC n ( B ) ≤ c ( n ) ̺ n , where one is allowed to chooseany positive constant c ( n ). The reason is that one uses HC ( r n +3 ) n in the proof ofLemma 3.6, and it is not clear what is the correct replacement of this quantity if r isallowed to be variable. So, we do not know how to prove a Hausdorff content analogof Theorem 1.1 where the radii of “small” balls can be variable. Acknowledgements.
This work was partially supported by NSERC Discoverygrant of the author.I would like to thank Roman Karasev who noticed several typos in the first ver-sion of the paper and suggested a simplification of the proof of Lemma 2.5, andalso for communicating to me the inequality sys ( X n ) ≤ U R n − ( X n ) for essentialpolyhedral length spaces. I would like to thank Anton Petrunin who asked me if theconstant in original version of Lemma 2.1 can be improved. This led me to a versionwith a better constant and helped to somewhat improve the constants in severaltheorems. References [BZ] Burago, Yu. D., Zalgaller, V., Geometric inequalities, Springer, 1988[Ga] Gaffney, M, “The conservation property of the heat equation on Riemannian manifolds”,Comm. Pure and Applied Math., 12(1959), 1-11.[Gr] Gromov, M., Filling Riemannian Manifolds, J. Differential Geom., 18(1983), 1-147.[Gu10] Guth, L., “Systolic inequalities and minimal hypersurfaces”, Geom. Functional Analysis(GAFA), 19(2010), 1688-1692.[Gu11] Guth, L., Volumes of balls in large Riemannian manifolds. Ann. of Math. (2) 173 (2011),no. 1, 51-76.6