Ball polytopes and the Vázsonyi problem

By Y. S. Kupitz, Horst Martini, M. Perles. 2009

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AbstractLet V be a finite set of points in the Euclidean d-space (d ≧ 2). The intersection of all unit balls B(υ, 1) centered at υ, where υ ranges over V, henceforth denoted by $$ \mathcal{B} $$(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) Define the boundary complex of $$ \mathcal{B} $$(V), i.e., define its vertices, edges and facets in dimension 3, and investigate its basic properties.(b) Apply results of this investigation to characterize finite 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 Vextremal if its diameter is attained this maximal number (2|V| − 2) of times. We extend the aforementioned result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope $$ \mathcal{B} $$(V) and show that in this case the boundary complex of $$ \mathcal{B} $$(V) is self-dual in some strong sense. The problem of constructing new types of extremal configurations is not addressed in this paper, but we do present here some such new types.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Discrete Mathematics and Combinatorics

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