I have to optimize a function $f$ over a set $S \subset X$. We know that $f$ is non-negative, continuos and strictly concave over $X$. We have that $S$ is compact but not convex.
By Extreme Value Theorem $f$ attains its maximum over $S$.
My question is this extremum unique?
I know that if $S$ is convex than the maximum is unique. However, in my case $S$ in not convex.
What further assumption on $f$ will guarantee uniqueness.