Convexity of Chebyshev Sets Through Differentiability of Distance Function

TL;DR

This paper investigates the convexity of Chebyshev sets in Banach spaces by examining the Gateaux differentiability of the distance function, providing conditions for convexity in approximation theory.

Contribution

It introduces a new approach linking the convexity of Chebyshev sets to the differentiability properties of the distance function in Banach spaces.

Findings

Convexity of Chebyshev sets is characterized by Gateaux differentiability of the distance function.

Provides new conditions under which Chebyshev sets are convex.

Enhances understanding of approximation theory in Banach spaces.

Abstract

In this paper, we study a part of approximation theory that presents the conditions under which a \Ceby\sev set in a Banach space is convex. To do so, we use Gateaux differentiability of the distance function.