Ball polytopes and the Vázsonyi problem

Abstract

Let V be a nite set of points in the Euclidean d-space (d 2).The intersection of all unit balls B(v, 1) centered at v, where v ranges over V , henceforth denoted by B(V ) is the ball polytope associated with V . After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes.(a) Dene the boundary complex of B(V ), i.e., dene its vertices, edges and facets in dimension 3, and investigate its basic properties. 2 (b) Apply results of this investigation to characterize nite sets of diameter 1 in the (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V . A basic result for such a characterization goes back to Grünbaum, Heppes and Straszewicz, who proved independently of each other, in the late 1950's by means of ball polytopes, that the diameter of V is attained at most 2|V | − 2 times. Call V extremal if its diameter is attained this maximal number 2|V | − 2 of times. We extend the aforementioned result by showing that V is extremal i V coincides with the set of vertices of its ball polytope B(V ) and show that in this case the boundary complex of B(V ) is self-dual in some strong sense. The problem of constructing 2 This face structure turns out to be quite intricate or, quoting [2], p. 202, lines 2223: the face structure of these objects [ball polytopes] is not obvious at all.

Dowker-type theorems for hyperconvex discs

“…for example Blanc [5], Buter [6], Pasqualini [21], Santaló [22], van der Corput [23], Vincensini [24], see also the survey paper by Danzer, Grünbaum, and Klee [8]), which, however, was forgotten by the turn of the century. More than seven decades after Mayer, Bezdek et al [4] and Kupitz et al [17,18] made a thorough investigation of such sets in E n (for more information see…”

“…Bezdek et al [4] and Kupitz et al [17,18] call such sets spindle convex, while L. Fejes Tóth [15,16], who proved packing and covering theorems for them, used the term r-convex. We use the English translation of Mayer's phrase.…”

On Random Disc Polygons in Smooth Convex Discs

“…Fejes Tóth proved packing and covering theorems for R-spindle convex discs in [15] and [16]. More recently, Bezdek et al [6] and Kupitz et al [20], [21] investigated spindle convex sets and proved numerous results about them, many of which are analogous to those of linearly convex sets. They also considered higher dimensional R-spindle convex sets.…”

“…Fejes Tóth in [15], [16] called such sets "R-convex". Bezdek et al [6] and Kupitz et al [20,21] used the expression "spindle convex". The notion of spindle convexity arose naturally and was investigated from different points of view, which explains the various names used for these sets and it also indicates their importance.…”

Globally Rigid Ball-Polyhedra in Euclidean 3-Space

“…This definition implies among others that any standard ball-polyhedron of E 3 is generated by at least four unit balls. For a number of important properties of ball-polyhedra we refer the interested reader to [4], [5], and [10].…”