On the number of vertices of projective polytopes

2018/10/05 by García-Colín, Natalia, Montejano, Luis Pedro, Alfonsín, Jorge Luis Ramírez · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.02671

Abstract

Let X be a configuration of n points in ℝ^d. What is the maximum number of vertices that conv(T(X)) can have among all the possible permissible projective transformations T? In this paper, we investigate this and connected questions. After presenting several upper bounds, we study a closely related problem (via Gale transforms) concerning the number of minimal Radon partitions of a set of points. We then present some bounds for this number that enable us to partially answer a question due to Pach and Szegedy. We also discuss another related problem concerning the size of topes in arrangements of hyperplanes.

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