Ball polytopes and the Vazsonyi problem
| dc.creator | Kupitz, Yaakov S. | |
| dc.creator | Martini, Horst | |
| dc.creator | Perles, Micha A. | |
| dc.date | 2009-05-11 | |
| dc.date.accessioned | 2026-07-07T13:13:37Z | |
| dc.date.available | 2026-07-07T13:13:37Z | |
| dc.description | Let V be a finite set of points in 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. Note that B(V) is non-empty iff the circumradius of V is <= 1. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. [a] Define the boundary complex of B(V) (assuming it is non-empty, of course), i.e., define its vertices, edges and facets in dimension 3 (in dimension 2 this complex is just a circuit), and investigate its basic properties. [b] Apply results of this investigation to characterize finite sets of diameter 1 in (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 Grunbaum, Heppes and Straszewicz, who proved independently that the diameter of V is attained at most 2|V|-2 times, thus affirming a conjecture of Vazsonyi from circa 1935. Call V extremal if its diameter is attained this maximal number (2|V|-2) of times. We extend the aforementioned basic result by showing that V is extremal iff 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. For the sake of priority we mention that, in the present form (except for a few changes in the footnotes), the paper was submitted to a journal already in February 1, 2008. | |
| dc.description | 80 pages, 15 figures | |
| dc.identifier | https://arxiv.org/abs/0905.1528 | |
| dc.identifier | http://arxiv.org/abs/0905.1528 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/229949 | |
| dc.subject | Metric Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | 52C10, 52A30, 52B10, 52B05, 05C62, 52A40 | |
| dc.title | Ball polytopes and the Vazsonyi problem | |
| dc.type | text |