Let $f: S \to \mathbb{R}$ be a (strictly) concave function, where
$S := \{y \in \mathbb{R}^m: y\geq 0,\, \sum_{i=1}^m y_i=1 \}$.
I want to show that there is a $y^*\in S$, which maximizes $f$. $S$ is a convex set and I found some results from convex optimization that strictly convex functions on convex sets have unique minimizers if local minima exist. Here, the set $S$ is also compact; how can we use this to prove the existence of a maximizing $y^*\in S$?
Thanks in advance!