Some stability properties of restricted Chebyshev centers in Banach spaces

TL;DR

This paper investigates the stability of Chebyshev centers in various Banach spaces, establishing existence results and stability properties for vector-valued function spaces and their Chebyshev-center maps.

Contribution

It proves the existence and stability of Chebyshev centers in function spaces like C(K,X) and explores their continuity properties, extending known results.

Findings

Existence of Chebyshev centers in C(K,X) for finite-dimensional X

Stability of restricted Chebyshev centers in vector-valued function spaces

Continuity properties of Chebyshev-center maps depend on the underlying space

Abstract

In this paper, we discuss the stability of (restricted) Chebyshev centers in few function spaces. For an extremally disconnected compact Hausdorff space $K$ and a finite dimensional Banach space $X$, we prove the existence of Chebyshev centers of closed bounded subsets of the space of $X$-valued continuous functions on $K$, $C(K,X)$ and that of compact subsets of $M$-ideals in $C(K,X)$. It is also proved that the existence of restricted Chebyshev centers is stable in the space of vector-valued bounded functions on an arbitrary set. Furthermore, the stability of continuity properties of the Chebyshev-center map of a Banach space $X$ in dependence on the continuity properties of the Chebyshev-center map of $C(K,X)$ is studied.