No-dimensional Helly's theorem in uniformly convex Banach spaces

TL;DR

This paper extends classical Helly-type theorems to uniformly convex Banach spaces using geometric inequalities related to the modulus of convexity, broadening the scope of these combinatorial geometric results.

Contribution

It introduces no-dimensional Helly-type theorems in uniformly convex Banach spaces, adapting proofs from Euclidean space with new geometric inequalities.

Findings

Helly's theorem holds in uniformly convex Banach spaces

Fractional and colorful Helly's theorems are established in this setting

Proofs rely on geometric inequalities linked to the modulus of convexity

Abstract

We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem, colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in \cite{adiprasito2020theorems}. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space.