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Licensed Unlicensed Requires Authentication Published by De Gruyter November 28, 2008

On reduced polytopes and antipodality

  • Gennadiy Averkov and Horst Martini
From the journal Advances in Geometry

Abstract

Let B be an o-symmetric convex body in ℝd, and Md be the normed space with unit ball B. The Md-thickness ΔB(K) of a convex body K ⊆ ℝd is the smallest possible Md-distance between two distinct parallel supporting hyperplanes of K. Furthermore, K is said to be Md-reduced if ΔB(K′) < ΔB(K) for every convex body K′ with K′ ⊆ K and K′ ≠ K. In our main theorems we describe Md-reduced polytopes as polytopes whose face lattices possess certain antipodality properties. As one of the consequences, we obtain that if the boundary of B is regular, then a d-polytope with m facets and n vertices is not Md-reduced provided m = d + 2 or n = d + 2 or n > m. The latter statement yields a new partial answer to Lassak's question on the existence of Euclidean reduced d-polytopes for d ≥ 3.

Received: 2007-03-13
Revised: 2007-09-05
Published Online: 2008-11-28
Published in Print: 2008-October

© de Gruyter 2008

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