Improved Tverberg theorems for certain families of polytopes

2024/04/17 by Soberón, Pablo, Zerbib, Shira · 2 citations
#52A37 #55M20 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.11533

Abstract

A theorem of Grünbaum, which states that every m-polytope is a refinement of an m-simplex, implies the following generalization of Tverberg's theorem: if f is a linear function from an m-dimensional polytope P to ℝ^d and m ≥ (d + 1)(r - 1), then there are r pairwise disjoint faces of P whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map f is continuous and r is a prime power. In this note, we show that for certain families of polytopes the lower bound on the dimension m of the polytopes can be significantly improved, both in the affine and topological cases.

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