Polyak-{\L}ojasiewicz inequality is essentially no more general than strong convexity for $C^2$ functions
Abstract
The Polyak-{\L}ojasiewicz (P{\L}) 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 P{\L} functions is rooted in the nonsmooth case since sufficient regularity forces them to be essentially strongly convex. More precisely, we prove that if is a P{\L} function having a bounded set of minimizers, then it has a unique minimizer and is strongly convex on a sublevel set of the form . 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.
Keywords
Cite
@article{arxiv.2512.05285,
title = {Polyak-{\L}ojasiewicz inequality is essentially no more general than strong convexity for $C^2$ functions},
author = {Aziz Ben Nejma},
journal= {arXiv preprint arXiv:2512.05285},
year = {2026}
}