Perturbation of farthest points in weakly compact sets

TL;DR

This paper investigates the density of points in a Banach space where certain supremum functions, involving weakly compact sets and weakly lower semi-continuous functions, are attained, revealing conditions and counterexamples.

Contribution

It establishes the density of points where a supremum involving a weakly lower semi-continuous function is attained and provides a counterexample for a related supremum involving norms.

Findings

Set of points where $z o orm{x-z} - f(z)$ attains supremum is dense in $X$.

Counterexample shows set where $z o orm{x-z} + orm{z}$ attains supremum is not always dense.

Highlights differences in supremum attainment for various functions on weakly compact sets.

Abstract

If $f$ is a real valued weakly lower semi-continous function on a Banach space $X$ and $C$ a weakly compact subset of $X$, we show that the set of $x \in X$ such that $z \mapsto \|x-z\|-f(z)$ attains its supremum on $C$ is dense in $X$. We also construct a counter example showing that the set of $x \in X$ such that $z \mapsto \|x-z\|+\|z\|$ attains its supremum on $C$ is not always dense in $X$.