# Square distance functions are Polyak-Lojasiewicz and vice-versa

arXiv: https://arxiv.org/abs/2301.10332 · HTML: https://mlsanity.com/paper/2301.10332/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2023.

## Built on by (most cited first, 3 of 3)

- [Distributionally Time-Varying Online Stochastic Optimization under Polyak-Lojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning](https://mlsanity.com/paper/2309.09411/) (2023)
- [From Inverse Optimization to Feasibility to ERM](https://mlsanity.com/paper/2402.17890/) (2024)
- [Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes](https://mlsanity.com/paper/2505.11259/) (2025)

## Abstract (arXiv)

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