Square distance functions are Polyak-Łojasiewicz and vice-versa

Abstract

This short note gathers known results to state that the squared distance function to a (nonconvex) closed set of an Euclidean space is Polyak-Łojasiewicz. As a fuzzy reciprocate, we also recall that every Polyak-Łojasiewicz function can be bounded from below by the squared distance function to its set of minimizers.

8 pages, 2 figures