On Chebyshev centers in Banach spaces

TL;DR

This paper investigates the property of Banach spaces where every finite set has a Chebyshev center, providing an answer to a previously posed problem and exploring the property’s behavior under certain space decompositions.

Contribution

It demonstrates that the property (GC) related to Chebyshev centers is not a 3-space property, extending understanding of geometric properties in Banach spaces.

Findings

Spaces with property (GC) admit Chebyshev centers for all finite sets.

The property (GC) is not preserved under certain space decompositions.

Answers a longstanding open problem in the geometry of Banach spaces.

Abstract

In this note, we observe that $X\in (GC)$ if $X$ admits Chebyshev center for any finite set in it. This answers the problem raised by Vesel\'y, addressed in [{\em Generalized centers of finite sets in Banach spaces}, Acta Math. Univ. Comenian. (N.S.) {\bf 66}(1) (1997), 83--115]. We extend our observation towards the fact that the property $(GC)$ is not a 3-space property.