No-dimension Tverberg's theorem and its corollaries in Banach spaces of type $p$

TL;DR

This paper extends no-dimension combinatorial and convex geometry theorems, including Tverberg's theorem, to Banach spaces of type p, using modified Maurey lemma techniques.

Contribution

It generalizes no-dimension versions of key theorems to Banach spaces of type p, expanding their applicability beyond Euclidean spaces.

Findings

No-dimension colorful Tverberg's theorem established in Banach spaces of type p.

Selection lemma and weak ε-net theorem extended to these Banach spaces.

Uses modified Maurey lemma to achieve these generalizations.

Abstract

We continue our study of 'no-dimension' analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper \cite{adiprasito2019theorems} and prove no-dimension versions of colorful Tverberg's theorem, selection lemma and the weak $\epsilon$-net theorem in Banach spaces of type $p > 1.$ To prove this results we use the original ideas of \cite{adiprasito2019theorems} for the Euclidean case and our slightly modified version of the celebrated Maurey lemma.