Quantitative Combinatorial Geometry for Continuous Parameters

Jesús A. De Loera, Reuben N. La Haye, David Rolnick, Pablo Soberón

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’s theorem, Bárány’s colorful Carathéodory’s theorem, and the colorful Tverberg’s theorem.