If $f$ is a real valued weakly lower semi-continuous 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 $X$.