Quantitative combinatorial geometry for continuous parameters

TL;DR

This paper develops continuous quantitative versions of classical combinatorial geometry theorems like Carathéodory's, Helly's, and Tverberg's, incorporating measures such as volume and diameter to extend their applicability.

Contribution

It introduces new continuous quantitative variants of key theorems, including colorful versions of Helly, Carathéodory, and Tverberg theorems, expanding their scope.

Findings

Proved continuous versions of Helly's, Carathéodory's, and Tverberg's theorems.

Established colorful quantitative theorems involving volume and diameter.

Extended classical combinatorial geometry results to measure-based settings.

Abstract

We prove variations of Carath\'eodory'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 Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful Carath\'eodory's theorem, and the colorful Tverberg theorem.