On remotality for convex sets in Banach spaces

TL;DR

This paper proves that in every infinite dimensional Banach space, there exists a closed, bounded convex set that is not remotal, highlighting a fundamental geometric property of such spaces.

Contribution

It establishes the existence of non-remotal convex sets in all infinite dimensional Banach spaces, a new insight into their geometric structure.

Findings

Existence of non-remotal convex sets in all infinite dimensional Banach spaces

Highlights geometric differences between finite and infinite dimensional spaces

Provides a new perspective on the structure of convex sets in Banach spaces

Abstract

We show that every infinite dimensional Banach space has a closed and bounded convex set that is not remotal.