The convex hull of a Banach-Saks set

TL;DR

This paper investigates whether the convex hull of a Banach-Saks set remains Banach-Saks, providing counterexamples and conditions under which the property is preserved.

Contribution

It demonstrates that the convex hull of a Banach-Saks set is not always Banach-Saks and offers sufficient conditions for the property to hold.

Findings

Counterexamples show the convex hull may not be Banach-Saks

Sufficient conditions ensure the convex hull is Banach-Saks

Provides combinatorial arguments for the analysis

Abstract

A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{\`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.