Extendability of Metric Segments in Gromov--Hausdorff Distance
aa r X i v : . [ m a t h . M G ] A ug Extendability of Metric Segments inGromov–Hausdorff Distance
S.I. Borzov, A.O. Ivanov, A.A. Tuzhilin
Abstract
In this paper geometry of Gromov–Hausdorff distance on the class ofall metric spaces considered up to an isometry is investigated. For thisclass continuous curves and their lengths are defined, and it is shown thatthe Gromov–Hausdorff distance is intrinsic. Besides, metric segments areconsidered, i.e., the classes of points lying between two given ones, and anextension problem of such segments beyond their end-points is considered.
Keywords:
Gromov–Hausdorff distance, class of all metric spaces,von Neumann–Bernays–Gödel axioms, intrinsic distance function, metricsegment, extendability of a segment beyond its end-points.
Introduction
In this paper the class of all metric spaces considered up to an isometry andendowed with the Gromov–Hausdorff distance is investigated. Recall that thisdistance is defined as a measure of “unlikeness” of metric spaces. In fact, thedistance is equal (up to multiplication by ) to the least possible distortion ofmetric over all correspondences (multivalued analogues of bijections) betweenthose spaces. As any distance, it permits to define a convergence of sequences,and as this convergence of metric spaces, so as the distance itself are activelyused in many different applications, such as, for example, groups growth veloc-ities or images recognition and comparison.If one restricts himself by compact metric spaces, then the Gromov–Hausdorffdistance is a metric. The resulting metric space M is referred as the Gromov–Hausdorff space, and many its properties are well-studied. For example, thisspace is path-connected, Polish (i.e., separable and complete), and geodesic.Also, the Gromov Criterion is well-known that gives a necessary and sufficientcondition for a subset of the Gromov–Hausdorff space to be pre-compact.Even a minimal extension of this space to the family of proper spaces leadsto new effects. First, the distance between different spaces can be infinite.Second, the distance between non-isometric spaces can be zero. But the triangleinequality for the Gromov–Hausdorff distance remains valid for arbitrary metricspaces.In the non-compact case, a modification of the Gromov–Hausdorff distancethat is referred as the pointed convergence is traditionally considered. Namely,1 . Preliminaries GH of all metric spaces considered up to an isometry.To work with such a “monster–space” we use the von Neumann–Bernays–Gödelaxioms system that permits to define correctly a distance function even on sucha proper class. But to generate a topology on this class by means of this distancefunction in the standard way turns out to be impossible (see below). We sug-gest a way to avoid this obstacle by means of a filtration over cardinality. As aresult, we define continuous curves in GH and show that the Gromov–Hausdorffdistance is an intrinsic extended pseudometric, i.e., the distance between anytwo points is equal to the infimum of the lengths of curves connecting thesepoints.The second part of the paper is devoted to geometry of metric segments in GH , where a metric segment is defined as a class of points lying between twofixed ones. We show that a metric segment could be a proper class, not a set(Conjecture: It is always the case). Besides, we study the possibility to extend ametric segment (to another one) beyond some its endpoint. This problem turnsout to be rather non-trivial, and its complete solution is unknown even forthe Gromov–Hausdorff space M . The key result is Theorem 3.19 (see below)that gives some sufficient condition of non-extendability of a metric segmentbeyond one of its endpoints. The final Section 4 contains many examples. Letus emphasize an interesting Example 4.16 based on Hadwiger Theorem solvingBorsuk Problem in a particular case. At the end of this Section it is also shownthat no metric segment, whose endpoints are bounded metric spaces, can beinfinitely extended beyond both its endpoints.The work is partly supported by RFBR (Project 19-01-00775-a) and by theMGU Program supporting scientific schools. Let X be an arbitrary set. By X we denote the cardinality of X , and let P ( X ) stand for the set of all non-empty subsets of X . A distance function ona set X is any symmetric mapping d : X × X → [0 , ∞ ] vanishing at the pairs ofcoinciding elements. If d satisfies the triangle inequality, then d is referred as a extended pseudometric . If in addition d ( x, y ) > for all x = y , then d is calledan extended metric . If d ( x, y ) < ∞ for all x, y ∈ X , then such distance functionis called a metric , and sometimes a finite metric to highlight the difference froman extended metric. A set X with an (extended) (pseudo-)metric is called an( extended ) ( pseudo- ) metric space . . Preliminaries X is a set with some distance function, then, as a rule, we denote thedistance between points x and y by | xy | . If γ is a curve in X , then by | γ | wedenote its length. Further, let x, y ∈ X and | xy | < ∞ , then we say that a point z ∈ X lies between x and y if | xz | + | zy | = | xy | . The set of all z lying between x and y is called the metric segment between x and y and is denoted by [ x, y ] .If | xy | > , then the metric segment is called non-degenerate . Definition 1.1.
We say that a metric segment [ x, y ] can be extended beyondthe point y if there exists a point z ∈ X such that [ x, z ] contains y and | yz | > . Remark 1.2. If | xy | = 0 , then any point z satisfies | zy | = | zx | = | zx | + | xy | = | zy | + | yx | , and if | zx | > , then z gives an extension of the metric segment [ x, y ] as beyond x , so as beyond y . We are mostly interested in extendability ofnon-degenerate metric segments.Let X be a metric space. For any A, B ∈ P ( X ) and x ∈ X we put | xA | = | Ax | = inf (cid:8) | xa | : a ∈ A (cid:9) , | AB | = inf (cid:8) | ab | : a ∈ A, b ∈ B (cid:9) ,d H ( A, B ) = max { sup a ∈ A | aB | , sup b ∈ B | Ab |} = max (cid:8) sup a ∈ A inf b ∈ B | ab | , sup b ∈ B inf a ∈ A | ba | (cid:9) . The function d H : P ( X ) × P ( X ) → [0 , ∞ ] is called the Hausdorff distance .It is well-known [1] that d H is a metric on the family H ( X ) ⊂ P ( X ) of allnon-empty closed bounded subsets of X .Let X and Y be metric spaces. A triplet ( X ′ , Y ′ , Z ) consisting of a met-ric space Z and two its subsets X ′ and Y ′ isometric to X and Y , respec-tively, is called a realization of the pair ( X, Y ) . The Gromov–Hausdorff distance d GH ( X, Y ) between X and Y is defined as the infimum of values r for whichthere exists a realization ( X ′ , Y ′ , Z ) of the pair ( X, Y ) such that d H ( X ′ , Y ′ ) ≤ r .Notice that the Gromov–Hausdorff distance could take as finite so as in-finite values, and it always satisfies the triangle inequality, see [1]. Besides,this distance equals zero for any pair of isometric spaces, therefore, due to thetriangle inequality, the Gromov–Hausdorff distance is well-defined on isometryclasses of metric spaces (it does not depend on a choice of representatives inthe classes). There are some examples of non-isometric metric spaces with zeroGromov–Hausdorff distance [4].Since any non-empty set can be endowed with some metric (for example, onecan put all non-zero distances to be equal to ), so there are “as many” isometryclasses of metric spaces as all possible sets, i.e., the family of isometry classes isnot a set, but it is a proper class which, together with the Gromov–Hausdorffdistance, is denoted by GH . Here we use the concept of class in the sense ofvon Neumann–Bernays–Gödel Set Theory (NBG). Recall some concepts of theNBG.In NBG all the objects (analogues of usual sets) are referred as classes .There are two types of classes: the sets and the proper classes . An example of aproper class is the class of all sets. According to the Gödel construction, one candistinguish a set from a proper class as follows: for a set there always exists aclass containing this set as an element. For a proper class there is no such a class. . Preliminaries distance function , the ( extended ) pseudo-metric , and the ( extended ) metric are defined for any class, as for a class thatis a set, so as for a class that is a proper one, because the products and themappings are defined for all classes. However, definitions of other structures onproper classes could face some difficulties. For example, if one tries to definea topology on some proper class C in the usual way, then C itself must be anelement of the topology, therefore, C must be a set, a contradiction.To avoid this problem, we consider a “filtration” of a class C by its subclasses C n each of which consists of all the elements from C , whose cardinality is at most n , where n is a cardinal. Recall that elements of any class are sets, and hence,the concept of cardinality is well-defined for them. The main examples for usare the class GH defined above, and the class B of all bounded metric spacesconsidered up to an isometry. Notice that for any cardinal n the subclasses GH n and B n are sets. Indeed, the family of all cardinals that do not exceed a givenone is a set, and for any fixed cardinal n the family of all isometry classes ofmetric spaces of cardinality n “can not be greater” than the set of all subsets of X × X × R , where X is an arbitrary set of cardinality n .We say that a class C is set-filtered if all its subclasses C n are sets. Evidently,if a class C is a set, then it is set-filtered.Thus, let C be a set-filtered class. We say that this class satisfies some prop-erties, if these properties are valid for each set C n . Let us give some examples. • Let a distance function on C be given, then it induces a “usual” distancefunction on each set C n . Thus, in each C n all concepts of metric geometryare defined (see above), for example open balls, and they are sets. Thelatter gives an opportunity to define the metric topology τ n on C n takingthose balls as a base of topology. Clear that for n ≤ m the topology τ n isinduced by the topology τ m . • Moreover, we say that a topology on C is defined, if for any cardinal n atopology τ n is defined on C n in such a way that the following consistencycondition holds: if n ≤ m , then τ n is the topology induced by τ m . • The presence of some topology on a class C makes it possible to define, forexample, continuous mappings from a topological space Z to the class C .Notice that, in accordance with NBG axioms, for an arbitrary mapping f : Z → C from the set Z to the class C , the image f ( Z ) is a set, allwhose elements are also some sets, and their union ∪ f ( Z ) is a set ofsome cardinality n . Therefore each element from f ( Z ) has a cardinalityat most n , and hence, f ( Z ) ⊂ C n . We call the mapping f continuous if f is continuous as a mapping from Z to the topological space C n . Theconsistency condition implies that for any m ≥ n the mapping f is a . Preliminaries Z to C m also, and for any k ≤ n such that f ( Z ) ⊂ C k the mapping f | C k is continuous too. • The above construction gives us an opportunity to define continuous curvesin a class C endowed with a topology. • Let a distance function and the corresponding topology be given on a class C . We say that this distance function is intrinsic if it satisfies the triangleinequality, and for any two elements x, y ∈ C such that | xy | < ∞ thedistance | xy | equals the infimum of lengths of curves connecting x and y .Below we show that the Gromov–Hausdorff distance is intrinsic as on theclass GH , so as on the class B .The most well-investigated subset of GH is the set of isometry classes ofcompact metric spaces. This set is called the Gromov–Hausdorff space and isoften denoted by M . It is well-known [1, 6] that the restriction of the Gromov–Hausdorff distance to M is a metric, and the metric space M is Polish andgeodesic.Notice that to simplify notation, it is convenient not to distinguish isometryclasses of metric spaces from their representatives. We have already used suchconvention, namely, we have defined B as the class of bounded metric spaces considered up to an isometry . Below we use this identification more than once,and write X ∈ GH having in mind that X is some specific metric space.As a rule, calculation of the Gromov–Hausdorff distance between specificmetric spaces is rather hard problem, and for today it is known for a few pairsof spaces, see for example [8]. The most useful tool for calculation of such kindis the following equivalent definition of the Gromov–Hausdorff distance. Recallthat a relation between sets X and Y is defined as a subset of their Cartesianproduct X × Y . So, P ( X × Y ) is the set of all non-empty relations between X and Y . Definition 1.3.
For any
X, Y ∈ GH and any σ ∈ P ( X × Y ) the value dis σ = sup n(cid:12)(cid:12) | xx ′ | − | yy ′ | (cid:12)(cid:12) : ( x, y ) , ( x ′ , y ′ ) ∈ σ o is called the distortion of the relation σ .A relation R ⊂ X × Y between sets X and Y is called a correspondence ifthe canonical projections π X : ( x, y ) x and π Y : ( x, y ) y are surjective on R . By R ( X, Y ) we denote the set of all correspondences between X and Y . Theorem 1.4 ([1]) . For any
X, Y ∈ GH the equality d GH ( X, Y ) = 12 inf (cid:8) dis R : R ∈ R ( X, Y ) (cid:9) holds. In what follows we need the following estimates on the Gromov–Hausdorffdistance that can be easily verified by means of Theorem 1.4. By ∆ we denotethe single-point metric space. . The Gromov–Hausdorff Distance is Intrinsic Assertion 1.5.
For any
X, Y ∈ GH the following relations are valid • d GH (∆ , X ) = diam X ; • d GH ( X, Y ) ≤ max { diam X, diam Y } ; • If either X or Y has a finite diameter, then (cid:12)(cid:12) diam X − diam Y (cid:12)(cid:12) ≤ d GH ( X, Y ) . Corollary 1.6. If X, Y ∈ B , then [ X, Y ] is defined and [ X, Y ] ⊂ B . For topological spaces X and Y , we consider X × Y as a topological spacewith the standard topology of Cartesian product. So, closed relations and closedcorrespondences are defined.A correspondence R ∈ R ( X, Y ) is called optimal if d GH ( X, Y ) = dis R . By R opt ( X, Y ) we denote the set of all optimal correspondences between X and Y ,and by R c opt ( X, Y ) we denote the subset of R opt ( X, Y ) consisting of all closedoptimal correspondences. Theorem 1.7 ([7, 3]) . For any
X, Y ∈ M there exists a closed optimal cor-respondence and also a realisation ( X ′ , Y ′ , Z ) of the pair ( X, Y ) which theGromov–Hausdorff distance between X and Y attained at. Theorem 1.8 ([7, 3]) . For any
X, Y ∈ M and any R ∈ R c opt ( X, Y ) the family R t , t ∈ [0 , , of compact metric spaces, where R = X , R = Y , and for t ∈ (0 , the space R t is the set R with the metric (cid:12)(cid:12) ( x, y ) , ( x ′ , y ′ ) (cid:12)(cid:12) t = (1 − t ) | xx ′ | + t | yy ′ | , is a shortest curve in M connecting X and Y , and the length of this curve equals d GH ( X, Y ) . We will also use the following notations. Let X be an arbitrary set and m acardinal such that < m ≤ X . By C m ( X ) we denote the family of all possiblecoverings of the set X by its m non-empty subsets, and by D m ( X ) we denotethe family of all partitions of X into m non-intersecting subsets. Obviously, D m ( X ) ⊂ C m ( X ) . If X is a metric space, then for any D = { X i } i ∈ I ∈ C m ( X ) we put diam D = sup i ∈ I diam X i , α ( D ) = inf (cid:8) | X i X j | : i = j (cid:9) and define the values d m ( X ) = inf D ∈D m ( X ) diam D, α m ( X ) = sup D ∈D m ( X ) α ( D ) . Let C be a set-filtered class endowed with a distance function that satisfies thetriangle inequality. Let x, y ∈ C , | xy | < ∞ , and γ be a curve in C connecting x and y . A curve γ is said to be an ε -shortest for x and y if ≤ | γ |− | xy | ≤ ε . It is . Metric Segments and their Extendability C that satisfies the triangle inequality isintrinsic if and only if for any pair of elements x, y from C such that | xy | < ∞ ,and for any ε > , there exists an ε -shortest curve connecting x and y . We usethis reasonings to prove the following Theorem. Theorem 2.1.
Let X and Y be arbitrary metric spaces such that d GH ( X, Y ) < ∞ . Let R ∈ R ( X, Y ) be an arbitrary correspondence such that dis R − d GH ( X, Y ) ≤ ε. Then the family R t , t ∈ [0 , , of metric spaces, where R = X , R = Y , andfor t ∈ (0 , the space R t is the set R endowed with the metric (cid:12)(cid:12) ( x, y ) , ( x ′ , y ′ ) (cid:12)(cid:12) t = (1 − t ) | xx ′ | + t | yy ′ | , is an ε -shortest curve in GH connecting X and Y . Moreover, if X and Y arebounded spaces, then all the spaces R t are also bounded, i.e., the curve R t is an ε -shortest curve in B .Proof. Put n = R and consider the following correspondences R X ⊂ X × R and R Y ⊂ R × Y between X and R t and between R t and Y , respectively: R X = (cid:8)(cid:0) x, ( x, y ) (cid:1) : x ∈ X, ( x, y ) ∈ R (cid:9) , R Y = (cid:8)(cid:0) ( x, y ) , y (cid:1) : y ∈ Y, ( x, y ) ∈ R (cid:9) . Then dis R X = t dis R and dis R Y = (1 − t ) dis R . Further, taking the identi-cal correspondence between the spaces R t and R s , where s, t ∈ (0 , , we get d GH ( R t , R s ) ≤ | t − s | dis R . Put γ ( t ) = R t . The above implies that γ isa continuous mapping from [0 , to GH n , i.e., γ is a continuous curve in theclass GH . Besides, | γ | ≤ dis R , therefore | γ | ≤ d GH ( X, Y ) + ε , and hence, γ is an ε -shortest curve for X and Y . It remains to notice that diam R t ≤ max { diam X, diam Y } , and so, if X, Y ∈ B , then R t ∈ B for all t as well. Corollary 2.2.
The Gromov–Hausdorff distance on the class GH is an intrinsicextended pseudometric. It is an intrinsic finite pseudometric on the class B . We call the ε -shortest curve constructed in Theorem 2.1 linear . Remark 2.3.
The proof of Theorem 2.1 implies that the linear ε -shortest curveconnecting metric spaces X and Y , d GH ( X, Y ) < ∞ , is a Lipschitz curve in GH ,and d GH ( X, Y ) + ε is its Lipschitz constant. We start with a description of simple properties of ε -shortest curves in an ex-tended pseudometric space C , where C is a set-filtered class. Lemma 3.1.
Let x, y ∈ C , | xy | < ∞ , γ be an ε -shortest curve connecting x and y , and w be an arbitrary point of the curve γ . Then the segments γ xw and γ wy of the curve γ between the points x , w and w , y , respectively, are ε -shortestcurves for their endpoints, and, moreover, the inequality | xw | + | wy | − | xy | ≤ ε holds. . Metric Segments and their Extendability Proof.
Indeed, by the definition of an ε -shortest curve, additivity of the lengthof a curve, and the triangle inequality we have: ε ≥ | γ | − | xy | = | γ xw | + | γ wy | − | xy | ≥ (cid:0) | γ xw | − | xw | (cid:1) + (cid:0) | γ wy | − | wy | (cid:1) . Notice that the expressions in parentheses are non-negative, therefore each ofthem does not exceed ε , and hence, the curves γ xw and γ wy are ε -shortest.Further, | xw | + | wy | ≤ | γ xw | + | γ wy | = | γ xy | ≤ | xy | + ε. Lemma is proved.Generally speaking, the union of two ε -shortest curves is not an ε -shortestone. However, the following result holds. Lemma 3.2.
Let x, y ∈ C , | xy | < ∞ , and a point w ∈ C lie between the points x and y . Then the union of an ε -shortest curve γ xw for x, w and a δ -shortestcurve γ wy for w, y , respectively, is an ( ε + δ ) -shortest curve for x, y .Proof. Indeed, denote by γ xy the union of the curves γ xw and γ wy . Then | xy | = | xw | + | wy | ≥ | γ xw | − ε + | γ wy | − δ = | γ xy | − ( ε + δ ) . Lemma is proved.Lemmas 3.1 and 3.2 imply the following estimate.
Corollary 3.3.
Let x, y ∈ C , | xy | < ∞ , and a point w ∈ C lie between thepoints x and y . Let γ xw and γ wy be ε -shortest curves connecting x, w and w, y , respectively. Then for any points p ∈ γ xw and q ∈ γ wy the inequality | py | + | yq | − | pq | ≤ ε holds. Let us list several elementary properties of metric segments.
Lemma 3.4.
For any x, y, z ∈ C we have • If y ∈ [ x, z ] , then [ x, y ] ⊂ [ x, z ] and [ y, z ] ⊂ [ x, z ] ; • If y lies between x and z , then y lies also between any points x ′ ∈ [ x, y ] and z ′ ∈ [ y, z ] . Remark 3.5.
If the space C is geodesic, then extendability of a metric segment [ x, y ] beyond y is equivalent to existence of a point z ∈ C such that any shortestcurve connecting x and y is contained in a shortest curve connecting x and z . Remark 3.6. If C is a proper class, then a metric segment could also be aproper class. As an example, consider the space GH and the metric segment [∆ , ∆ n ] in it, where ∆ n stands for the metric space of cardinality n all whosenon-zero distances are equal to . It is easy to verify that the curve γ ( t ) , t ∈ [0 , , where γ (0) = ∆ and γ ( t ) = t ∆ n for all other t , is a shortest curveconnecting ∆ and ∆ n . Consider the space Z = ∆ n , choose its arbitrary . Metric Segments and their Extendability z ∈ Z , and change it with a non-empty set A of an arbitrary cardinality.Fix a positive ε < / . On the resulting set Z ′ = (cid:0) Z \ { z } (cid:1) ⊔ A redefine thedistance as follows: | aa ′ | = ε and | az ′ | = | zz ′ | = 1 / for any distinct a, a ′ ∈ A and any z ′ ∈ Z \ { z } . It is easy to verify that d GH ( Z ′ , ∆ ) = d GH ( Z, ∆ ) and d GH ( Z ′ , ∆ n ) = d GH ( Z, ∆ n ) , therefore, any such space Z ′ lies between ∆ and ∆ n , i.e., it belongs to the metric segment [∆ , ∆ n ] . Since the cardinality of thespace Z ′ is arbitrary, then [∆ , ∆ n ] is a proper class.Let X ∈ B , and C = { X i } i ∈ I be a covering of X by non-empty sets. We saythat { X i } i ∈ I is a covering by sets of less diameter than X if diam C < diam X . Lemma 3.7.
Let X ∈ B , and C X = { X i } i ∈ I be a covering of X by non-emptysubsets of less diameter than X . Put δ X = diam X − diam C X > . Then any Y ∈ B such that d GH ( X, Y ) < ε = δ X / , has a similar representation, i.e.,there exists a covering C Y = { Y i } i ∈ I of the space Y by non-empty subsets ofless diameter, and diam Y − diam C Y > ε .Proof. Since d GH ( X, Y ) < ε , then there exists an R ∈ R ( X, Y ) such that dis R < ε . For each i ∈ I put Y i = R ( X i ) . Since R is a correspondence, then Y i = ∅ for any i ∈ I , and C Y := { Y i } i ∈ I is a covering of Y by non-empty sets.Since dis R < ε , diam Y i ≤ diam X i + dis R , and diam X ≤ diam Y + dis R ,then diam C Y ≤ diam C X + dis R < diam C X + 2 ε = diam X − δ X + 2 ε == diam X − ε ≤ diam Y + dis R − ε < diam Y − ε. Lemma is proved.
Lemma 3.8.
Let X ∈ GH , and C X = { X i } i ∈ I be a covering of X by non-emptysubsets of finite diameter. Then any Y ∈ GH such that d GH ( X, Y ) < ∞ has asimilar representation, i.e., there exists a covering C Y = { Y i } i ∈ I of the space Y by non-empty subsets of finite diameter, and diam C Y ≤ diam C X +2 d GH ( X, Y ) .Proof. Let d GH ( X, Y ) < ε , then there exists an R ∈ R ( X, Y ) such that dis R < ε . For each i ∈ I put Y i = R ( X i ) . Since R is a correspondence, then Y i = ∅ for any i ∈ I , and C Y := { Y i } i ∈ I is a covering of Y by non-empty sets. Since dis R < ε , then diam Y i ≤ diam X i + dis R and diam C Y ≤ diam C X + 2 ε , thatimplies the statement of Lemma.By means of the following standard construction, one can pass form a cov-ering to a partition whose cardinality and diameter do not increase. Construction 3.9.
Let X ∈ GH , and C = { X i } i ∈ I be a covering of X suchthat diam C < diam X . By well-ordering Zermelo’s theorem, we can introducea strict total order on the index set I , and put X ′ i = X i \ [ j : j
Let
X, Y ∈ B be such that (1) d := diam Y − diam X > ;(2) There exists a partition D X = { X i } i ∈ I of the space X with α ( D X ) > ;(3) There exists a covering C Y = { Y j } j ∈ J of the space Y by subsets of lessdiameter such that δ Y := diam Y − diam C Y > ;(4) J ≤ I .Then diam Y − d GH ( X, Y ) ≥ min { d, α ( D X ) , δ Y } > .Proof. Using Construction 3.9 described above, reconstruct the covering C Y = { Y j } of the space Y into a partition D Y = { Y ′ k } k ∈ K , K ≤ J ≤ I , diam D Y ≤ diam C Y . Let σ : K → I be an arbitrary injection. Consider thefollowing correspondence: R = (cid:16) [ k ∈ K \{ k } ( X σ ( k ) × Y ′ k ) (cid:17) [(cid:16) [ i ∈ I \ σ ( K ) ( X i × Y ′ k ) (cid:17) , where k ∈ K is any fixed element. Estimate the distortion of the correspon-dence R . Since elements form distinct Y ′ k always correspond in R to elementsfrom distinct X i , then d GH ( X, Y ) ≤ dis R ≤ max (cid:8) diam X, diam Y − α ( D X ) , diam D Y (cid:9) ≤≤ max (cid:8) diam Y − d, diam Y − α ( D X ) , diam Y − δ Y (cid:9) ≤≤ diam Y − min { d, α ( D ) , δ Y } . Lemma is proved.
Definition 3.11.
Let
X, Y ∈ B be such that d GH ( X, Y ) = 0 . We say that Y is hyperextreme with respect to X if d GH ( X, Y ) = diam Y ≥ diam X, and that Y is subextreme with respect to X if d GH ( X, Y ) = diam Y − diam X. Remark 3.12. If Y is hyperextreme with respect to X , then, due to Asser-tion 1.5, the distance d GH ( X, Y ) between X and Y takes the greatest possi-ble value for the spaces with such diameters. Conversely, if d GH ( X, Y ) =max { diam X, diam Y } > and diam Y ≥ diam X , then Y is hyperextreme withrespect to X . Remark 3.13. If Y is subextreme with respect to X , then diam Y > diam X and, due to Assertion 1.5, the distance d GH ( X, Y ) takes the least possible valuefor the spaces X and Y with such diameters. Conversely, if d GH ( X, Y ) = | diam X − diam Y | > , then the space with a larger diameter is subextremewith respect to the one with a smaller diameter. . Metric Segments and their Extendability Remark 3.14.
Any metric space Y ∈ B , Y = ∆ , is simultaneously hyperex-treme and subextreme with respect to the single-point space ∆ . This is theonly such case: if Y is simultaneously subextreme and hyperextreme with re-spect to some X , then d GH ( X, Y ) = diam Y = diam Y − diam X , therefore diam X = 0 , and hence, X = ∆ . Definition 3.15. If d GH ( X, Y ) = diam Y = diam X > , then Y is hyperextreme with respect to X , and X is hyperextreme with respectto Y . In this case, we say that the spaces X and Y are mutually hyperextreme ,and the metric segment [ X, Y ] ⊂ B is said to be extreme . Remark 3.16.
Assertion 1.5 implies that the spaces X and Y are mutuallyhyperextreme if and only if they are “diametrally opposite” points of the sphereof radius diam X centered at the single-point space ∆ , i.e., X and Y arepoints of this sphere most distant from each other. To the contrary, if Y issubextreme with respect to X , then Y is the closest to X point of the sphere ofradius diam Y centered at ∆ , i.e., X and Y “lie at the same radial ray”. Assertion 3.17.
Let
X, Y ∈ B , and the metric segment [ X, Y ] be extreme.Assume that [ X, Y ] can be extended beyond Y to some Z . Then Z ∈ B , diam Z > diam Y , and the space Z is subextreme with respect to the space Y .Proof. Indeed, since Y lies between X and Z due to assumptions, then, by def-inition, d GH ( X, Z ) < ∞ , and hence, Z ∈ B in accordance with Assertion 1.5.Moreover, d GH ( X, Z ) = d GH ( X, Y ) + d GH ( Y, Z ) , and d GH ( Y, Z ) > by defi-nition of the extendability, so d GH ( X, Z ) > d GH ( X, Y ) . On the other hand, if diam Z ≤ diam Y , then, in accordance with Assertion 1.5, we have d GH ( X, Z ) ≤ max { diam X, diam Z } = diam X, and diam X = 2 d GH ( X, Y ) because the segment [ X, Y ] is extreme, and so d GH ( X, Y ) = d GH ( X, Z ) , a contradiction. Thus, diam Z > diam Y .Further, using the assumptions that Y lies between X and Z , that thesegment [ X, Y ] is extreme, and Assertion 1.5, we have: max (cid:8) diam X, diam Z (cid:9) ≥ d GH ( X, Z ) = 2 d GH ( X, Y ) + 2 d GH ( Y, Z ) == diam X + 2 d GH ( Y, Z ) ≥ diam X + (cid:12)(cid:12) diam Y − diam Z (cid:12)(cid:12) . However, since diam Z ≥ diam Y = diam X , then the leftmost part and therightmost part of this relations chain equal to diam Z , so d GH ( Y, Z ) = diam Z − diam Y > . Lemma 3.18.
Let
Y, Z ∈ B , and Z be subextreme with respect to Y . Fixan arbitrary ε from the interval (0 , diam Z − diam Y ) and consider a linear ε -shortest curve R t . Then diam R t ≥ (1 − t ) diam Y + t diam Z − ε. . Metric Segments and their Extendability Proof.
Let R ∈ R ( Y, Z ) be an arbitrary correspondence such that dis R ≤ d GH ( Y, Z ) + 2 ε . Choose z, z ′ ∈ Z such that | zz ′ | ≥ diam Z − ε , and hence, | zz ′ | > diam Y . Then for any y ∈ R − ( z ) and y ′ ∈ R − ( z ′ ) we have: (cid:12)(cid:12) | zz ′ | − | yy ′ | (cid:12)(cid:12) ≤ dis R ≤ d GH ( Y, Z ) + 2 ε = diam Z − diam Y + 2 ε, where the latter equality holds because Z is subextreme with respect to Y . Onthe other hand, (cid:12)(cid:12) | zz ′ | − | yy ′ | (cid:12)(cid:12) = | zz ′ | − | yy ′ | ≥ diam Z − ε − | yy ′ | due to thechoice of z, z ′ and ε , therefore, diam Z − diam Y + 2 ε ≥ diam Z − ε − | yy ′ | , and hence, diam Y − | yy ′ | ≤ ε . Thus, diam R t ≥ (cid:12)(cid:12) ( y, z ) , ( y ′ , z ′ ) (cid:12)(cid:12) t = (1 − t ) | yy ′ | + t | zz ′ | ≥≥ (1 − t )(diam Y − ε ) + t (diam Z − ε ) ≥≥ (1 − t ) diam Y + t diam Z − ε. Lemma is proved.
Theorem 3.19.
Let
X, Y ∈ B , m and n be cardinal numbers such that < n ≤ X , < m ≤ Y . Suppose that the metric segment [ X, Y ] is extreme and thefollowing conditions hold :(1) There exists a partition D X ∈ D n ( X ) such that α ( D X ) > ;(2) There exists a covering C Y ∈ C m ( Y ) , such that diam C Y < diam Y ;(3) m ≤ n .Then the metric segment [ X, Y ] can not be extended beyond Y .Proof. To the contrary assume that there exist a metric space Z such that Y lies between X and Z . Due to Assertion 3.17, in this case Z ∈ B and Z issubextreme with respect to Y . The following Lemma is evident. Lemma 3.20.
For any δ > and d > there exists an ε > such that theinequality εd < δ d + ε ) (1) holds for all ε ∈ [0 , ε ) . Under notations of Lemma 3.20, choose the corresponding ε for δ = diam Y − diam C Y and d = d GH ( Y, Z ) = diam Z − diam Y > . Fix an arbitrary ε suchthat < ε < min (cid:26) ε , d , δ , α ( D X )4 (cid:27) , . Metric Segments and their Extendability ε < d/ , then the lefthand side of Inequality (1) is less than , therefore there exists an s ∈ (0 , such that εd < s < δ d + ε ) . Consider any R ∈ R ( Y, Z ) with dis R ≤ d + 2 ε and the corresponding linear ε -shortest curve R t , t ∈ [0 , , connecting Y and Z , where R = Y , R = Z .For the s chosen above, consider the space R s and by reasoning similar to theproof of Theorem 2.1, we get that d GH ( Y, R s ) ≤ s dis R , and so, d GH ( Y, R s ) ≤ s (2 d + 2 ε ) < δ d + ε ) (2 d + 2 ε ) = δ/ . Due to Lemma 3.7, there exists a covering C R of the space R s by non-emptysubsets of less diameter, such that C R = C Y and diam R s − diam C R > δ/ .Since diam Y < diam Z and ε < s d , than, by Lemma 3.18, we get diam R s ≥ diam Y + s (diam Z − diam Y ) − ε = diam Y + s d − ε > diam Y + 4 ε, and so, because diam X = diam Y , we have diam R s − diam X > ε > . Thelatter, together with the above estimates, imply that the pair X and R s satisfiesall conditions of Lemma 3.10, and so, d GH ( X, R s ) ≤≤ diam R s − min (cid:8) diam R s − diam X, α ( D ) , diam R s − diam C R (cid:9) . However, each of the three expressions standing under the minimum in the previ-ous formula are strictly greater than ε , and hence, d GH ( X, R s ) < diam R s − ε .On the other hand, d GH ( Y, R s ) ≥ (cid:12)(cid:12) diam Y − diam R s | = diam R s − diam Y ,and so, applying Corollary 3.3 and taking into account the condition d GH ( X, Y ) =diam Y , we obtain that d GH ( X, R s ) ≥ d GH ( X, Y ) + 2 d GH ( Y, R s ) − ε ≥≥ d GH ( X, Y ) + diam R s − diam Y − ε = diam R s − ε, a contradiction. Theorem is proved. Remark 3.21.
One might get the impression that the statement of Theo-rem 3.19 can be weaken by changing the mutual hyperextremality of the space X and Y by hyperextremality of Y with respect to X . But if Y is hyperex-treme with respect to X and Conditions (1) – (3) of Theorem 3.19 are valid,then the spaces X and Y are mutually extreme. Indeed, if diam Y > diam X ,then Lemma 3.10 implies the inequality d GH ( X, Y ) < diam Y that contra-dicts to hyperextremality of Y with respect to X , and hence only the case d GH ( X, Y ) = diam Y = diam X is possible, but it is the case of mutual ex-tremality. Corollary 3.22.
Let
X, Y ∈ M , the spaces X and Y be mutual hyperextreme,and Conditions (1) − (3) of Theorem . hold. Then any shortest curve in M connecting X and Y can not be extended beyond Y . . Some Examples Above, by ∆ we denoted the single-point metric space, and λ ∆ n was the met-ric space of cardinality n , all whose non-zero distances are equal to λ . Suchspaces are called one distance spaces or simplexes for brevity. Some formulasfor calculating the Gromov–Hausdorff distances from the simplexes to boundedmetric spaces can be found in [8]. We use these results to construct examplesof metric segments that can be extended. Let X be a metric space and m a cardinal number that does not exceed X .Recall that we have already defined several characteristics of possible partitionsof a metric space X into m subsets. Theorem 4.1 ([8]) . For X ∈ B and a cardinal number < m ≤ X theequality d GH ( λ ∆ m , X ) = inf D ∈D m max (cid:8) diam D, λ − α ( D ) , diam X − λ (cid:9) is valid. Here we need a particular case of this formula.
Corollary 4.2.
For X ∈ B , a cardinal number < m ≤ X , and λ ≥ diam X + α m ( X ) the equality d GH ( λ ∆ m , X ) = λ − α m ( X ) holds.Proof. Indeed, if λ ≥ diam X + α m ( X ) , then λ − α ( D ) ≥ λ − α m ( X ) ≥ diam X ≥ diam D and hence, λ − α ( D ) ≥ diam X − λ . Therefore, for such λ the maximumin the formula from Theorem 4.1 equals λ − α ( D ) , so d GH ( λ ∆ m , X ) = inf D ( λ − α ( D )) = λ − sup D α ( D ) = λ − α m ( X ) . Corollary is proved. Corollary 4.3.
For X ∈ B , a cardinal number < m ≤ X , and λ ≥ diam X + α m ( X ) the metric segment [ X, λ ∆ m ] can be extended beyond λ ∆ m to any simplex λ ′ ∆ m , where λ ′ > λ .Proof. Indeed, since d GH ( λ ∆ m , λ ′ ∆ m ) = | λ − λ ′ | and in accordance with Corol-lary 4.2, for any λ ≥ diam X + α m ( X ) the equality d GH ( λ ∆ m , X ) = λ − α m ( X ) holds. Therefore, the simplex λ ∆ m lies between X and λ ′ ∆ m , where λ ′ > λ , asrequired. The next Assertion formalizes Remark 3.16.
Assertion 4.4.
Let
X, Y ∈ B , and Y be subextreme with respect to X . Then X lies between ∆ and Y . . Some Examples Proof.
Indeed, d GH (∆ , Y ) = diam Y = diam X + (cid:0) diam Y − diam X (cid:1) == 2 d GH (∆ , X ) + 2 d GH ( X, Y ) , that is required. Corollary 4.5.
Let
X, Y ∈ B , and Y be subextreme with respect to X , and X = ∆ . Then the metric segment [ X, Y ] is extendable as beyond X , so asbeyond Y . Construction 4.6.
Let Y ∈ B and there exist points y , y ∈ Y such that diam Y = | y y | . For arbitrary r ≥ and r ≥ , construct a two-pointextension Z r ,r ( y , y ) of the space Y as follows. Put Z = Z r ,r ( y , y ) = Y ⊔ { z , z } and extend the metric from Y to Z by the next rule: | z z | = r + | y y | + r , and for y ∈ Y put | yz i | = r i + | yy i | , i = 1 , . If r i = 0 , thenidentify z i with y i . It is easy to verify that Z r ,r ( y , y ) is a metric space forany r ≥ and r ≥ , and diam Z r ,r ( y , y ) = diam Y + r + r = | z z | . The spaces Z r ( y ) := Z r , ( y , y ) and Z r ( y ) := Z ,r ( y , y ) we call a single-point extensions of the space Y . Clear that Z , ( y , y ) = Z ( y ) = Z ( y ) = Y. Lemma 4.7.
Let Y ∈ B and there exist points y , y ∈ Y such that diam Y = | y y | . Let Z r ,r ( y , y ) be a two-point extension of the space Y constructedabove. Then for the curve γ ( t ) = Z r t,r t ( y , y ) , t ∈ [0 , , the equality d GH (cid:0) γ ( t ) , γ ( t ) (cid:1) = | t − t | ( r + r ) holds for all t , t ∈ [0 , , and hence, γ is a shortest curve connecting Y and Z r ,r ( y , y ) . In particular, putting r = 0 or r = 0 , one gets a shortest curveconnecting Y with the corresponding single-point extension Z r ( y ) or Z r ( y ) .Proof. Consider the correspondence R ∈ R (cid:0) γ ( t ) , γ ( t ) (cid:1) that is identical on theset Y ∪ { z , z } . Then dis R = max (cid:8) | t − t | r , | t − t | r , | t − t | ( r + r ) (cid:9) = | t − t | ( r + r ) , and so, d GH (cid:0) γ ( t ) , γ ( t ) (cid:1) ≤ | t − t | ( r + r ) . On the other hand, (cid:12)(cid:12) diam γ ( t ) − diam γ ( t ) (cid:12)(cid:12) == (cid:12)(cid:12) diam Y + ( r + r ) t − diam Y − ( r + r ) t (cid:12)(cid:12) = | t − t | ( r + r ) , . Some Examples d GH (cid:0) γ ( t ) , γ ( t ) (cid:1) ≥ | t − t | ( r + r ) ,so d GH (cid:0) γ ( t ) , γ ( t ) (cid:1) = | t − t | ( r + r ) , that implies Lemma’s statement. Assertion 4.8.
Let
X, Y ∈ B , and Y be subextreme with respect to X . Assumethat there exist y , y ∈ Y such that diam Y = | y y | . Then the metric segment [ X, Y ] is extendable beyond Y to any two-point extension Z := Z r ,r ( y , y ) ofthe space Y .Proof. By Lemma 4.7, we have d GH ( Y, Z ) = r + r .Now, estimate d GH ( X, Z ) by means of Assertion 1.5 and the triangle in-equality: diam Z − diam X ≤ d GH ( X, Z ) ≤ d GH ( X, Y ) + 2 d GH ( Y, Z ) == diam Y − diam X + r + r = diam Z − diam X, therefore, d GH ( X, Z ) = diam Z − diam X == (cid:0) diam Z − diam Y (cid:1) + (cid:0) diam Y − diam X (cid:1) == 2 d GH ( Y, Z ) + 2 d GH ( X, Y ) , that is required. Remark 4.9.
In Assertion 4.8, the metric segment [ X, Y ] can be extendedbeyond Y in different ways: as to Z r ( y ) , so as to Z r ( y ) , and to Z r ,r ( y , y ) as well. Therefore, if say X, Y ∈ M , where any two points are connected bya shortest curve, than any shortest curve connecting X and Y can branch outat the point Y , similarly to the situation when a geodesic comes to the vertexof a flat cone with total angle at this vertex more than π . In particular, thissituation takes place for X = ∆ and the standard “radial” shortest curve of theform t t Y , t ∈ [0 , . Remark 4.10.
If we restrict ourselves by the space M of compact metricspaces, then for any X, Y ∈ M that satisfy the conditions of Assertion 4.8,and for any r > , any shortest curve connecting X and Y can be extendedbeyond Y to any space Z from the intersection of the sphere S (∆ ) of radius / Y + r/ centered at ∆ with the sphere S ( Y ) of radius r/ centeredat Y . In particular, this intersection contains all single-point extensions of thespace Y with the parameter r , and all its two-point extensions with parameters r , r , where r + r = r .The proof of the next Lemma can be obtained by a trivial modification ofthe proof of Lemma 4.7. . Some Examples Lemma 4.11.
Assume that Y ∈ B and there exist y , y ∈ Y such that diam Y = | y y | . Let Z r ,r ( y , y ) be a two-point extension of the space Y defined above. Then for any ≤ s ≤ r and ≤ s ≤ r the equality d GH (cid:0) Z r ,r ( y , y ) , Z s ,s ( y , y ) (cid:1) = s − r + s − r holds. Lemma 4.11 implies the following result.
Assertion 4.12.
Let Y and Z r ,r ( y , y ) be such as in Lemma . , and r ( t ) and s ( t ) be non-negative strictly increasing continuous functions on t ∈ I , where I is a finite or infinite interval. Then the curve γ ( t ) = Z r ( t ) ,s ( t ) ( y , y ) isshortest. By means of Theorem 3.19, one can construct examples of non-extendable metricsegments and non-extendable shortest curves.
Example 4.13.
Let X = λ ∆ n and Y = λ ∆ m be single-distance spaces, and < m < n . Then the metric segment [ X, Y ] can not be extended beyond Y .Indeed, d GH ( X, Y ) = λ = diam X = diam Y , therefore the spaces X and Y are mutually extreme with respect to each other. Further, X and Y canbe partitioned into n and m single-point subsets of zero diameter, respectively,therefore for m < n we are under assumptions of Theorem 3.19.We need another simple corollary from Theorem 4.1, see [8]. Corollary 4.14.
Let X be a bounded metric space, < m ≤ X , and α m ( X ) =0 . Then d GH ( λ ∆ m , X ) = max (cid:8) d m ( X ) , λ, diam X − λ (cid:9) . Example 4.15.
Now, let X = ∆ be the single-distance space consisting of twopoints with non-zero distance , and let Y = [0 , be the segment of the length .Then α ( Y ) = 0 , d ( Y ) = 1 / , and hence d GH (∆ , Y ) = 1 in accordance withCorollary 4.14. As we have already seen in Corollary 4.3, the metric segment [ X, Y ] can be extended beyond the simplex X . On the other hand, the spaces X and Y are mutually extreme. Represent the simplex X as the union of itstwo points, and the segment Y as the union of two segments Y = [0 , / and Y = [1 / , . Theorem 3.19 implies that [ X, Y ] can not be extended beyond Y . Example 4.16.
Example 4.15 can be generalized as follows. Let X = ∆ k bethe unit simplex consisting of k points, < k < ∞ , and Y be a closed convexbody of diameter with a smooth boundary in ( k − -dimensional Euclideanspace. Then α k ( Y ) = 0 , and the metric segment [ X, Y ] can be extended beyondthe simplex X . Due to Hadwiger’s theorem [9], see also [10], the space Y canbe covered by k subsets of diameters less than , therefore d k ( Y ) < , and so, d GH (∆ k , Y ) = 1 in accordance with Corollary 4.14. So, the spaces X and Y are mutually extreme. Represent the simplex X as the union of its points, andcover Y as above by k subsets of diameters less than . Then Theorem 3.19 canbe applied, and the segment [ X, Y ] can not be extended beyond Y . eferences In this Section we show that no one of metric segments in the space B canbe infinitely extended to both sides. We start with the following Lemma. Lemma 4.17.
There is no metric segments [ X , X ] ⊂ GH of the length s inthe interior of the ball of radius s centered at the single-point space ∆ .Proof. Due to Assertion 1.5, we have s ≤ max { diam X , diam X } , and since d GH (∆ , X i ) = diam X i , then the distance from ∆ to one of X i is more orequal than s . Lemma is proved. Assertion 4.18.
Let Z ∈ B be an interior point of an ε -shortest curve γ .Then at least one of the ends of the curve γ is contained in the ball of radius R = diam Z + ε centered at the single-point space ∆ .Proof. Let an ε -shortest curve γ connect points X and X , and both thesepoints lie outside the ball of radius R centered at ∆ . Then the segment of thecurve γ between Z and X i , i = 1 , , contains a point Y i lying at the sphere S of a radius R ′ > R centered at ∆ . Therefore, diam Y i = 2 R ′ , i = 1 , , and d GH ( Y i , Z ) ≥ diam Y i − diam Z = 2 R ′ − diam Z , i = 1 , . Further, the segment γ between Y and Y is also an ε -shortest curve. Therefore, due to Lemma 3.1, d GH ( Y , Y ) ≥ d GH ( Y , Z ) + d GH ( Z, Y ) − ε ≥ R ′ − diam Z − ε. On the other hand, d GH ( Y , Y ) ≤ max { diam Y , diam Y } = R ′ , and hence, R < R ′ ≤ diam Z + ε , a contradiction. Corollary 4.19.
No metric segment can be infinitely extended beyond both itsendpoints.Proof.
Assume to the contrary that for an arbitrary ε > and each i = 1 , there exists an extension of some metric segment [ X , X ] beyond its end X i tosome Y i such that d GH ( X i , Y i ) > max { diam X , diam X } + ε . Due to extensiondefinition, Y i ∈ B , i = 1 , . In accordance to Corollary 2.2, there exist ( ε/ -shortest curves connecting Y and X , X and X , and X and Y , respectively.By Lemma 3.2, the union of these three curves is an ε -shortest curve between Y and Y , and it contains X and X in its (relative) interior. By Assertion 4.18,one of the end-points of this curve, i.e., one of the points Y i , say Y , is containedin the ball of radius max { diam X , diam X } + ε centered at ∆ , i.e., diam Y ≤ { diam X , diam X } + 2 ε . By Assertion 1.5, we have d GH ( Y , X ) ≤
12 max { diam Y , diam X } ≤ max { diam X , diam X } + ε, a contradiction. eferences References [1] D. Burago, Yu. Burago, and S. Ivanov,