Gromovemdash.cyrHausdorﬀ distance between vertex sets of regular polygons inscribed in a given circle Talant Talipov

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Gromov—Hausdorﬀ distance between vertex sets of

regular polygons inscribed in a given circle

Talant Talipov

Abstract

We calculate the Gromov–Hausdorﬀ distance between vertex sets of regular polygons

endowed with the round metric. We give a full answer for the case of n- and m-gons

with mdivisible by n. Also, we calculate all distances to 2-gons and 3-gons

1 Introduction

In the present paper we study the class of all metric spaces considered up to isometry,

endowed with the Gromov–Hausdorﬀ distance. Note that the exact values of the Gromov–

Hausdorﬀ distances between speciﬁc metric spaces are known only for a small number of

cases. For example, in [1], the Gromov—Hausdorﬀ distances from a broad class of metric

spaces to 1-spaces, i.e. metric spaces with one non-zero distance, were calculated. In [2],

the Gromov—Hausdorﬀ distance between a segment and a circle with intrinsic metric was

obtained. In [3], the Gromov—Hausdorﬀ distances between spheres of diﬀerent dimensions

were calculated in some cases and estimated in other ones. Also, the Gromov–Hausdorﬀ

distance between the vertex set of regular polygons and circle were calculated, as well as

between diﬀerent regular m- and (m+ 1)-gons inscribed in the same circle. In the present

paper we extend the results of [3] for the case of m- and n-gons, provided that mis divisible

by n. Also we have calculated all distances to 2-gons and 3-gons. The author expresses his

gratitude to his supervisor, Prof. Alexey A. Tuzhilin, and to Prof. Alexander O. Ivanov for

posing the problem and help in the work.

2 Preliminaries

Let Xbe an arbitrary metric space. The distance between points x, y we denote by d(x, y)

or |xy|. For any non-empty A,Bthe Hausdorﬀ distance between Aand Bis deﬁned as

follows :

dH(A, B) = maxsup

a∈A

inf

b∈Bd(a, b),sup

b∈B

inf

a∈Ad(a, b).

Let Xand Ybe metric spaces. If X0and Y0are subsets of a metric space Z0, provided X0

is isometric to X, and Y0is isometric to Y, then (X0, Y 0, Z0)is called realization of the pair

arXiv:2210.09971v1 [math.MG] 18 Oct 2022

(X, Y ).The Gromov–Hausdorﬀ distance between Xand Yis the value

dGH (X, Y ) = infr:∃(X0, Y 0, Z0), dH(X0, Y 0)6r.

Deﬁnition 2.1. Given two sets Xand Y, a correspondence between Xand Yis a subset

R⊂X×Ysuch that for any x∈Xthere exists y∈Ywith (x, y)∈Rand, vise versa, for

any y∈Ythere exists x∈Xwith (x, y)∈R. If X,Yare metric spaces, then we deﬁne the

correspondence distortion Ras follows : dis R= supn|x1x2|−|y1y2|: (x1, y1),(x2, y2)∈Ro.

We denote by R(X, Y )the set of all correspondences between Xand Y.

Theorem 2.1 ([4]).Let Xand Ybe metric spaces. Then

dGH (X, Y ) = 1

2infdis R:R∈ R(X, Y ).

In [3] the following construction was considered. For a metric space (X, dX), we deﬁne the

pseudo-ultrametric space (X, uX)where uX:X×X→R+is deﬁned by

uX: (x, y)7→ infmax

06i6n−1dX(xi, xi+1) : x0=x, ..., xn=y.

Now, deﬁne U(X)to be the quotient metric space of (X, uX)over the equivalence x∼yif

and only if uX(x, y) = 0.

Theorem 2.2 ([3]).For all bounded metric spaces Xand Y, it holds

dGH (X, Y )>dGH U(X),U(Y).

Deﬁnition 2.2. By simplex we call a metric space, all whose non-zero distances equal to each

other. If mis an arbitrary cardinal number, then by λ∆mwe denote a simplex containing

mpoints and such that all its non-zero distances equal to λ.

Let Xbe an arbitrary set consisting of more than one point, 26m6#Xa cardinal number,

and λ > 0. By Dm(X)we denote the family of all possible partitions of the set X into m

non-empty subsets. For any non-empty A, B ⊂X, we put |AB|= inf|ab|:a∈A, b ∈B.

Now let Xbe a metric space. Then for each D={Xi}i∈I∈ Dm(X)we put

diam D= sup

i∈I

diam Xi;

α(D) = inf|XiXj|:i6=j.

Theorem 2.3 ([1]).Let X6= ∆1be an arbitrary metric space, 26m6#Xa cardinal

number, and λ > 0. Then

2dGH (λ∆, X) = inf

D∈Dm(X)maxdiam D, λ −α(D),diam X−λ.

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摘要：

GromovHausdordistancebetweenvertexsetsofregularpolygonsinscribedinagivencircleTalantTalipovAbstractWecalculatetheGromovHausdordistancebetweenvertexsetsofregularpolygonsendowedwiththeroundmetric.Wegiveafullanswerforthecaseofn-andm-gonswithmdivisiblebyn.Also,wecalculatealldistancesto2-gonsand3-gon...

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