aa r X i v : . [ m a t h . GN ] S e p Metric Segments in Gromov–Hausdorff class.
Borisova Olga Borisovna
Student, Lomonosov State University, Faculty of Mechanics and Mathematics
Abstract
We study properties of metric segments in the class of all metricspaces considered up to an isometry, endowed with Gromov–Hausdorffdistance. On the isometry classes of all compact metric spaces, theGromov-Hausdorff distance is a metric. A metric segment is a classthat consists of points lying between two given ones. By von Neumann–Bernays–G¨odel (NBG) axiomatic set theory, a proper class is a “mon-ster collection”, e.g., the collection of all cardinal sets. We prove thatany metric segment in the proper class of isometry classes of all me-tric spaces with the Gromov-Hausdorff distance is a proper class ifthe segment contains at least one metric space at positive distancesfrom the segment endpoints. In addition, we show that the restrictionof a non-degenerated metric segment to compact metric spaces is anon-compact set.
Keywords:
Gromov–Hausdorff distance, class of all metric spaces,von Neumann–Bernays–G¨odel axioms, metric segment
The Gromov-Hausdorff distance is a measure of difference between two ar-bitrary metric spaces. It is closely related with Hausdorff distance. TheHausdorff distance first appeared in the book of Hausdorff entitled “Set the-ory” [1] in 1914. This value defines a natural distance on non-empty subsetsof a metric space and generates a metric on the set of all non-empty closedbounded subsets.In 1981, Gromov in his monograph [2] introduced a distance betweenarbitrary metric spaces: he embedded isometrically two metric spaces intocommon metric spaces and considered the Hausdorff distances between theirimages. The infimum of the possible Hausdorff distances over all such em-beddings is called the Gromov-Hausdorff distance.1t is well-known that this distance satisfies the triangle inequality andvanishes for any isometric metric spaces. It is correctly to consider theGromov-Hausdorff distance on isometry classes of metric spaces, which canbe calculated on arbitrary representatives of the classes (it does not dependon the choice of the representatives). Since any non-empty set can be en-dowed with some metric (for example, one can put all non-zero distancesto be equal to 1), all isometry classes do not form a set due to the famousCantor’s paradox. In order to overcome this problem, we use in this paperthe von Neumann–Bernays–G¨odel (NBG) axiomatic set theory [3].In NBG, all the objects (analogues of usual sets) are referred as classes.A class is a set if there exists a class containing this one as an element,otherwise, the class is called a proper class. The class of all sets is a properclass. For the classes many standard operations are defined, for example,products, mappings, etc. Therefore, on proper classes, similarly with thecase of sets, a distance is defined correctly. The class of isometry classes ofall metric spaces (as mentioned above) is a proper class, and being endowedwith the Gromov-Hausdorff distance, it is an extended pseudometric space,which we denote by GH .In this paper, we investigate the class of elements lying between twogiven points in extended pdeudometric space GH . A metric space Z ∈ GH lies between X, Y ∈ GH if | XZ | + | ZY | = | XY | , where | · ·| denotes theGromov-Hausdorff distance. The class of all such Z is called a metric segmentand is denoted by [ X, Y ] . We prove constructively, that if a metric segmentcontains Z ∈ GH at positive distances from endpoints, then this space Z canbe modified by adding a set with an arbitrary cardinal number such that thenew metric space remains in the segment. Therefore, such a metric segmentin GH is a proper class.The Gromov-Hausdorff distance is well studied on the set of all compactmetric spaces considered up to isometry, where it is a metric. The set of allisometry classes of compact metric spaces is called the Gromov-Hausdorffspace . In 2015, A. Ivanov, N. Nikolaeva, A. Tuzhilin showed that in Gromov-Hausdorff space any two points are joined by geodesics [4]. Therefore, anynon-degenerated segment is a non-empty set. The previous result leads tonon-compactness of the metric segments in the Gromov–Hausdorff space.I would like to express my gratitude to my advisor Dr. Sc. (Phys.-Math.)professor Tuzhilin A.A. and also to Dr. Sc. (Phys.-Math.) professor IvanovA.O. for permanent support of my study.The work is partly supported by RFBR (Project №19-01-00775а).2
Preliminaries
Let X be an arbitrary set. A distance function on X is any symmetric map-ping d : X × X → [0 , ∞ ] vanishing at the pairs of coinciding elements. If d satisfies the triangle inequality, then the mapping d is called an extendedpseudometric . If in addition d ( x, y ) > for all x = y , the d is referred asan extended metric . If also d ( x, y ) < ∞ for all x, y ∈ X , then such distancefunction is called a metric , and sometimes a finite metric to highlight thedifference from an extended metric. A set X with an (extended) (pseudo-)metric is called an ( extended ) ( pseudo- ) metric space .Let X be an arbitrary metric space. The distance between points x, y ∈ X we denote by | xy | . Sometimes, we add subscript | xy | X to highlight thelocation of the points. For any x ∈ X and non-empty A ⊂ X , we put | xA | = inf (cid:8) | xa | : a ∈ A (cid:9) . Let P ( X ) stand for the set of all non-emptysubsets of X . For any A, B ∈ P ( X ) let us put d H ( A, B ) = max (cid:8) sup a ∈ A | aB | , sup b ∈ B | Ab | (cid:9) . The value d H ( A, B ) is called the Hausdorff distance between A and B .Let B ( X ) ⊂ P ( X ) be the family of all non-empty bounded subsets of X . Proposition 1 ([5]) . The function d H ( A, B ) is a pseudometric on B ( X ) . By H ( X ) ⊂ B ( X ) we denote the family of all non-empty closed boundedsubsets of a metric space X . Proposition 2 ([5]) . The function d H ( A, B ) is a metric on H ( X ) . Let X and Y be metric spaces. A triple ( X ′ , Y ′ , Z ) consisting of a metricspace Z and its subsets X ′ and Y ′ isometric to X and Y , respectively, iscalled a realization of the pair ( X, Y ) . For any metric spaces X, Y , the value d GH ( X, Y ) = inf (cid:8) r : ∃ ( X ′ , Y ′ , Z ) , d H ( X ′ , Y ′ ) ≤ r (cid:9) is called the Gromov–Hausdorff distance between X and Y . Proposition 3 ([5]) . The function d GH ( X, Y ) satisfies the triangle inequa-lity. If X and Y are isometric metric spaces, then d GH ( A, B ) = 0 . Notice, that the Gromov–Hausdorff distance could take infinite values,and there are some examples of non-isometric metric spaces
X, Y for which d GH ( X, Y ) = 0 , see [5]. So, the Gromov–Hausdorff distance is well-defined3n isometry classes of metric spaces (it does not depend on the choice ofrepresentatives in the classes).Since any non-empty set can be endowed with some metric (for exam-ple, one can put all non-zero distances to be equal to 1), so there are “asmany” isometry classes of metric spaces as all possible sets, i.e., the familyof isometry classes is not a set, but it is a proper class which, together withthe Gromov–Hausdorff distance, is denoted by GH . Here we use the conceptof class in the sense of von Neumann–Bernays–G¨odel Set Theory (NBG).Recall some concepts of the NBG.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 . A class A iscalled a set if there exists a class C such that A ∈ C . A class A is called a proper class if for any class C it holds A 6∈ C . Proposition 4 ([3]) . The class of all sets V = {A : A = A} is a properclass. For all classes X , Y it is defined X × X , f : X → Y , etc., in particular, wecan speak about distance function on a class. Therefore, the distance on GH is defined correctly. The class of isometry classes of bounded metric spaces isa proper class too, and being endowed with the Gromov-Hausdorff distance,it is denoted by B . Proposition 5 ([5]) . The function d GH ( A, B ) is a pseudometric on B . The most well-investigated subset of GH is the set M of isometry classesof compact metric spaces. Proposition 6 ([5]) . The function d GH ( A, B ) is a metric on M . Now we describe a convenient way to calculate the Gromov–Hausdorffdistance. Let X and Y be non-empty sets. Put P ( X, Y ) = P ( X × Y ) . Eachelement of P ( X, Y ) is called a relation between X and Y .Let π X : X × Y → X and π Y : X × Y → Y be the canonical projections π X ( x, y ) = x and π Y ( x, y ) = y . A relation σ ∈ P ( X, Y ) is called a corre-spondence , if the restrictions of π X and π Y onto σ are surjections. The setof all correspondences between X and Y is denoted by R ( X, Y ) .For any metric spaces X and Y and for any relation σ ∈ P ( X, Y ) , wedefine its distortion as follows: dis σ = sup n(cid:12)(cid:12) | xx ′ | − | yy ′ | (cid:12)(cid:12) : ( x, y ) , ( x ′ , y ′ ) ∈ σ o . roposition 7 ([5]) . Let X and Y be metric spaces. Then d GH ( X, Y ) = 12 inf (cid:8) dis R : R ∈ R ( X, Y ) (cid:9) . A correspondence R ∈ R ( X, Y ) is called optimal if d GH ( X, Y ) = dis R .The set of all optimal correspondences between X and Y we denote by R opt ( X, Y ) . Let R c ( X, Y ) be the set of all correspondences closed in X × Y . Proposition 8 ([6]) . For
X, Y ∈ M it holds R opt ( X, Y ) ∩ R c ( X, Y ) = ∅ . Let
X, Y be arbitrary metric spaces. For each t ∈ (0 , we define adistance function on X × Y as (cid:12)(cid:12) ( x, y )( x ′ , y ′ ) (cid:12)(cid:12) t = (1 − t ) | xx ′ | + t | yy ′ | . Proposition 9 ([6]) . The function defined above | · | t is a metric for any t ∈ (0 , on X × Y . For any σ ∈ P ( X, Y ) , the metric space (cid:0) σ, | · | t (cid:1) is denoted by σ t . Proposition 10 ([6]) . For any
X, Y ∈ M and any closed σ ∈ P ( X, Y ) , themetric space σ t belongs to M for all t ∈ (0 , . Let us choose an arbitrary R ∈ R ( X, Y ) and extend the family R t , t ∈ (0 , , up to t = 0 , , as R = X and R = Y . Proposition 11 ([6]) . For any
X, Y ∈ M and R ∈ R opt ( X, Y ) ∩ R c ( X, Y ) ,the mapping t R t , t ∈ [0 , , is a curve connecting X and Y . This curve isa shortest curve, and its length equals d GH ( X, Y ) . Therefore, the metric ofthe space M is geodesic. Let X be a metric space and ε > . A covering number cov(
X, ε ) is theminimum number of opened balls with radius ε one needs in order to cover X . The following proposition is called the precompact theorem of Gromov .For metric space X we denote the diameter of space as diam X . Proposition 12 ([5]) . Let C be a non-empty subset M . Then the followingstatements are equivalent. (1) There exists a number D ≥ and a function N : (0 , ∞ ) → N such thatfor any X ∈ C the inequalities diam X ≤ D and cov( X, ε ) ≤ N ( ε ) arevalid. (2) The family C ⊂ M is precompact. A class X we call a simplex if all its non-zero distances are equal. For anycardinal n , we denote by λ ∆ n the simplex with n vertices and with distance λ between any its distinct points. For λ = 1 the space λ ∆ n we denote by ∆ n for short. 5 Main results If X is a class with some distance function, then a metric segment with ends A, B ∈ X is a class [ A, B ] = (cid:8) C ∈ X : | AC | + | CB | = | AB | (cid:9) . If | AB | > ,then the segment [ A, B ] is called non-degenerate .A closed ball in X with center x ∈ X and radius r > we denote by B r ( x ) = (cid:8) x ∈ X : | xx | ≤ r (cid:9) . Theorem 1.
Let a segment [ X, Y ] be non-degenerate for X, Y ∈ GH . Ifthere exists a metric space Z ∈ GH lying in [ X, Y ] such that d GH ( X, Z ) > and d GH ( Z, Y ) > , then the segment [ X, Y ] contains a space Z ∗ ∈ GH with at least one isolated point, and the conditions d GH ( X, Z ∗ ) > and d GH ( Z ∗ , Y ) > hold. Proof.
Suppose that Z does not contain isolated points, otherwise thetheorem is proved. Fix an arbitrary point z ∈ Z and a finite number δ suchthat < δ ≤ min (cid:8) d GH ( X, Z ) , d GH ( Y, Z ) (cid:9) . We construct a metric space Z ∗ = Z ∪ { z ∗ } preserving the distances betweenpoints z ∈ Z and assigning the distances from z ∈ Z ∗ to z ∗ as follows: | z ∗ z | Z ∗ = δ for z ∈ Z ∩ B δ ( z ) , | z z | Z for z ∈ Z \ B δ ( z ) , z = z ∗ . Make sure that this function is a metric. It is sufficient to check thetriangle inequality for z , z , z ∗ , where z , z ∈ Z , because other axioms areobvious.Consider different cases of the locations of points z , z .Let z , z ∈ Z ∩ B δ ( z ) then the triangle is isosceles and | z ∗ z | Z ∗ ≤ | z ∗ z | Z ∗ + | z z | Z ∗ , | z z | Z ∗ = | z z | Z ≤ | z z | Z + | z z | Z ≤ δ = | z z ∗ | Z ∗ + | z ∗ z | Z ∗ . If z , z ∈ Z \ B δ ( z ) , then the distances between z , z , z ∗ are equal tothe distances in Z between z , z , z , respectively. In the last case if z ∈ Z \ B δ ( z ) and z ∈ Z ∩ B δ ( z ) , then | z ∗ z | Z ∗ = | z z | Z ≤ | z z | Z + | z z | Z ≤ | z ∗ z | Z ∗ + | z z | Z ∗ , | z ∗ z | Z ∗ = δ ≤ | z ∗ z | Z ∗ ≤ | z ∗ z | Z ∗ + | z z | Z ∗ , z z | Z ∗ = | z z | Z ≤ | z z | Z + | z z | Z ≤ | z z ∗ | Z ∗ + | z ∗ z | Z ∗ . Therefore Z ∗ ∈ B .The next step is to prove d GH ( X, Z ∗ ) ≤ d GH ( X, Z ) .Consider an arbitrary correspondence R ∈ R ( X, Z ) and construct a newcorrespondence R ∗ = R ∪ (cid:0) R − ( z ) × { z ∗ } (cid:1) ∈ R ( X, Z ∗ ) , where R − ( z ) = (cid:8) x ∈ X : ( x, z ) ∈ R (cid:9) . The distortion of R ∗ consists of three parts: r = sup n(cid:12)(cid:12) | xx ′ | − | zz ′ | (cid:12)(cid:12) : ( x, z ) , ( x ′ , z ′ ) ∈ R ∗ ; z, z ′ ∈ Z o ,r = sup n(cid:12)(cid:12) | xx ′ | − | zz ∗ | (cid:12)(cid:12) : | zz | > δ ; ( x, z ) ∈ R ∗ ; x ′ ∈ R − ( z ) o ,r = sup n(cid:12)(cid:12) | xx ′ | − | zz ∗ | (cid:12)(cid:12) : | zz | ≤ δ ; ( x, z ) ∈ R ∗ ; x ′ ∈ R − ( z ) o , i.e, dis R ∗ = max { r , r , r } . Estimate r i from above.By construction of correspondence R ∗ , we have r = dis R .For r , the inequality | zz | > δ holds, then | zz ∗ | = | zz | and r = sup n(cid:12)(cid:12) | xx ′ | − | zz | (cid:12)(cid:12) : | zz | > δ ; ( x, z ) , ( x ′ , z ) ∈ R o ≤ dis R. Consider any points x, x ′ , z from the definition of r . If ≤ | xx ′ | − | zz ∗ | , then | xx ′ | − | zz ∗ | ≤ | xx ′ | − | zz | ≤ dis R. Otherwise, < | zz ∗ | − | xx ′ | ≤ δ ≤ d GH ( X, Z ) ≤ dis R. Hence, r ≤ dis R , and we get dis R ∗ ≤ dis R .Using this inequality and Proposition 7, we obtain d GH ( X, Z ∗ ) ≤ d GH ( X, Z ) . The inequality d GH ( Z, Y ) ≥ d GH ( Z ∗ , Y ) can be proved similarly. Then d GH ( X, Y ) = d GH ( X, Z )+ d GH ( Z, Y ) ≥ d GH ( X, Z ∗ )+ d GH ( Z ∗ , Y ) ≥ d GH ( X, Y ) . The last inequality is the triangle inequality for the Gromov–Hausdorff di-stance. Thus, d GH ( X, Z ∗ ) + d GH ( Z ∗ , Y ) = d GH ( X, Y ) , which means Z ∗ ∈ [ X, Y ] . Theorem 2.
Let a segment [ X, Y ] be non-degenerate for X, Y ∈ GH , andthere is a metric space Z ∈ [ X, Y ] with at least one isolated point such that d GH ( X, Z ) > , d GH ( Z, Y ) > . Then for an arbitrary cardinal m there is ametric space W ( m ) ∈ GH lying in the segment [ X, Y ] , where W ( m ) containsa simplex with m vertices. If Z ∈ M and the cardinal m is a finite, then W ( m ) ∈ M . roof. Let z ∗ ∈ Z be an isolated point. Fix a finite number µ such that < µ < (cid:8) d GH ( X, Z ) , d GH ( Z, Y ) , S ( z ∗ ) (cid:9) , where S ( z ∗ ) = inf (cid:8) | z ∗ z | : z ∈ Z, z = z ∗ (cid:9) > . Let m be an arbitrary cardinalnumber. On the set W = W ( µ, m ) = µ ∆ m ∪ Z \ { z ∗ } we define the distance function as follows. Let w , w ∈ W , then | w w | W = | w w | Z for w , w ∈ Z \{ z ∗ } , | w i z ∗ | Z for w i ∈ Z, w − i ∈ µ ∆ m , i ∈ { , } ,µ for w , w ∈ µ ∆ m , w = w , otherwise.This function is a metric. The properties of positive definiteness and sym-metricity are obvious. We need to check the triangle inequality for differentpoints lying in W . If all three points w , w , w lie in Z \{ z ∗ } , or if w i ∈ µ ∆ m ,where i ∈ { , , } , then the triangle inequality holds . If w , w ∈ Z \{ z ∗ } ,and w ∈ µ ∆ m , then the distances between this points are equal to thedistances between w , w , z ∗ ∈ Z , respectively, so the triangle inequality isvalid in this case. If w ∈ Z \{ z ∗ } , and w , w ∈ µ ∆ m , then | w w | W = µ < S ( z ∗ ) ≤ | z ∗ w | Z = | w w | W + | w w | W , | w w | W = | w z ∗ | Z < | w z ∗ | Z + µ = | w w | W + | w w | W . Similarly for | w w | W . Therefore, W is a metric space.Now we prove that d GH ( X, W ) ≤ d GH ( X, Z ) .Consider an arbitrary correspondence R ∈ R ( X, Z ) and construct a newcorrespondence V ∈ R ( X, W ) as follows. Put ( x, w ) ∈ V if and only if w ∈ Z \ { z ∗ } and ( x, w ) ∈ R or w ∈ µ ∆ m and ( x, z ∗ ) ∈ R .Calculate the distortion of V : dis V = sup n(cid:12)(cid:12) | xx ′ | − | ww ′ | (cid:12)(cid:12) : ( x, w ) , ( x ′ , w ′ ) ∈ V o . The distortion of V consists of two parts: v = sup n(cid:12)(cid:12) | xx ′ | − | ww ′ | (cid:12)(cid:12) : ( x, w ) , ( x ′ , w ′ ) ∈ V ; w ∈ Z \ { z ∗ } , w ′ ∈ W o ,v = sup n(cid:12)(cid:12) | xx ′ | − | ww ′ | (cid:12)(cid:12) : ( x, w ) , ( x ′ , w ′ ) ∈ V ; w, w ′ ∈ µ ∆ m o , dis V = max { v , v } . For any pairs ( x, w ) , ( x ′ , w ′ ) ∈ V from the v ,there are pairs ( x, z ) , ( x ′ , z ′ ) ∈ R with z, z ′ ∈ Z and | ww ′ | W = | zz ′ | Z , thus, v ≤ dis R . Estimate the value v from above. Rewrite v in equivalent formtaking into account that | ww ′ | W = µ for w, w ′ ∈ µ ∆ m , and therefore v = sup n(cid:12)(cid:12) | xx ′ | − µ (cid:12)(cid:12) : ( x, z ∗ ) , ( x ′ , z ∗ ) ∈ R o . Due to the restriction on µ , we have µ < d GH ( X, Z ) ≤ dis R. In addition, for x, x ′ ∈ X such that ( x, z ∗ ) , ( x ′ , z ∗ ) ∈ R we have | xx ′ | ≤ dis R .Since the absolute value of difference between µ and | xx ′ | is less or equal than dis R , the value v is less or equal than dis R .So we proved dis V = max { v , v } ≤ dis R .According to this inequality and Proposition 7, we have d GH ( X, W ) ≤ d GH ( X, Z ) .Similarly, we obtain the inequality d GH ( Z, Y ) ≥ d GH ( W, Y ) . Then d GH ( X, Y ) = d GH ( X, Z )+ d GH ( Z, Y ) ≥ d GH ( X, W )+ d GH ( W, Y ) ≥ d GH ( X, Y ) , where the last inequality is the triangle inequality for the Gromov-Hausdorffdistance.Thus, d GH ( X, W ) + d GH ( W, Y ) = d GH ( X, Y ) , which means W ( µ, m ) ∈ [ X, Y ] for any cardinal m .If Z is a compact metric space, and m is a finite number, then the metricspace W ( m ) is a compact metric, because when constructing W ( m ) , we addonly a finite number of points to compact metric space. Corollary 1.
If for space
X, Y ∈ GH the metric segment [ X, Y ] contains ametric space Z ∈ GH such that d GH ( X, Z ) > and d GH ( Y, Z ) > , then themetric segment [ X, Y ] is a proper class. Proof.
According to Theorem 1 and Theorem 2, the metric segment [ X, Y ] can be surjectively mapped to the proper class of all cardinality sets.For instance, any space W ( m ) goes to a set with cardinal number m , andother elements of the segment go to single-point set. Therefore, the segment“is not less” than a proper class, so the segment is a proper class. Corollary 2.
For two different metric space
X, Y ∈ M , the segment [ X, Y ] is not compact. roof. By Proposition 11, the set [ X, Y ] \ { X, Y } is non-empty, thusit contains some Z . Space M is a metric space thus d GH ( X, Z ) > and d GH ( Z, Y ) > . According to Theorem 1 and Theorem 2 there exists acompact metric space W ( m ) ∈ [ X, Y ] for any positive integer m and W ( m ) contains an m -vertices simplex. Let the distance between differen points inthe simplex be µ > . For any < ε < µ the covering number cov (cid:0) W ( m ) , ε (cid:1) is not less than m , because each ball with radius ε can cover no more thanone point from µ ∆ m ⊂ W ( m ) . Therefore, for (cid:8) W ( m ) (cid:9) ∞ m =1 ⊂ [ X, Y ] thereis no function N : (0 , ∞ ) → N such that for any m ∈ N the inequality cov (cid:0) W ( m ) , ε (cid:1) ≤ N ( ε ) is valid. So, according to Proposition 12, the segment[X, Y] is not precompact and, hence, a compact set. Literatur [1] F. Hausdorff,
Grundz¨uge der Mengenlehre . Leipzig: von Veit, 1914.[2] M. Gromov,
Structures m´eriques pour les vari´et´es riemanniennes , editedby Lafontaine and Pierre Pansu, 1981.[3] E. Mendelson,
An Introduction to Mathematical Logic . 1997[4] A. Ivanov, N. Nikolaeva, A. Tuzhilin,
The Gromov-Hausdorff Me-tric on the Space of Compact Metric Spaces is Strictly Intrinsic ,arXiv:1504.03830 (2015).[5] D. Burago, Yu. Burago, and S. Ivanov,
A Course in Metric Geometry ,Graduate Studies in Mathematics, vol 33 (A.M.S., Providence, RI, 2001;Russian Edition: IKI, Moscow, Izhevsk, 2004).[6] Ivanov A.O., Iliadis S., Tuzhilin A.A.,