Quantitative combinatorial geometry for continuous parameters

Abstract

We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász's colorful Helly theorem, Bárány's colorful Carathéodory's theorem, and the colorful Tverberg theorem.

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

References in corpus (6)

• Optimal bounds for a colorful Tverberg--Vrecica type problem
• 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 Tverberg theorems over lattices and other discrete sets

Cited by in corpus (1)

• Quantitative Tverberg theorems over lattices and other discrete sets