Perturbation of farthest points in 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 $z\mapsto \Vert x-z\Vert -f(z)$ 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 $z\mapsto \Vert x-z\Vert +\Vert z\Vert$ attains its supremum on $C$ is not always dense in $X$.

arXiv comment5 pages