On Chebyshev centers in Banach spaces
Syamantak Das, Tanmoy Paul

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 if 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 is not a 3-space property.
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Taxonomy
TopicsAdvanced Banach Space Theory · Fixed Point Theorems Analysis · Nonlinear Differential Equations Analysis
