Polyak-Łojasiewicz inequality is essentially no more general than strong convexity for $C^2$ functions

The Polyak-Łojasiewicz (PŁ) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in an earlier blog post) showing that the richness of the class of PŁ functions is rooted in the nonsmooth case since sufficient regularity forces them to be essentially strongly convex. More precisely, we prove that if $f$ is a $C^2$ PŁ function having a bounded set of minimizers, then it has a unique minimizer and is strongly convex on a sublevel set of the form $\{f\leq a\}$. We show that this implies a result of Asplund on properties of the squared distance function, and discuss some consequences on smoothness assumptions in results in the literature.