Perturbation of farthest points in weakly compact sets

Details Perturbation of farthest points in weakly compact sets Perturbation of farthest points in weakly compact sets

2012-04-10

arxiv.org/abs/1204.2047v1

Mathematics

Functional Analysis

5 pages

Scientific paper

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 $X$.

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