Differentiability of Distance Function and The Proximinal Condition implying Convexity

TL;DR

This paper provides a new characterization of when the distance function to a closed set in a normed space is differentiable, linking it to convexity of Chebyshev sets without requiring uniform differentiability of the norm.

Contribution

It establishes a necessary and sufficient condition for the differentiability of the distance function under a proximinality condition, advancing previous results by relaxing norm differentiability assumptions.

Findings

Proximinal condition applies to almost suns.

Differentiability of the distance function implies convexity of certain sets.

Characterization of convex Chebyshev sets in Banach spaces with rotund dual.

Abstract

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles [16]. Hence, our result advances that of Giles [16]. We prove that the proximinal condition of Giles [16] is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev sets in Banach spaces with rotund dual.