Quantitative Tverberg theorems over lattices and other discrete sets

Abstract

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carathéodory's theorems.

arXiv comment16 pages. arXiv admin note: substantial text overlap with arXiv:1503.06116

References in corpus (5)

• Counterexamples to the topological Tverberg conjecture
• Quantitative Tverberg, Helly, & Carathéodory theorems
• Quantitative $(p,q)$ theorems in combinatorial geometry
• A Quantitative Doignon-Bell-Scarf Theorem
• Quantitative combinatorial geometry for continuous parameters

Cited by in corpus (1)

• Quantitative combinatorial geometry for continuous parameters