# Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers

arXiv: https://arxiv.org/abs/1507.07144 · HTML: https://mlsanity.com/paper/1507.07144/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2015.

## Built on by (most cited first, 5 of 5)

- [Convergence of First-Order Algorithms for Meta-Learning with Moreau Envelopes](https://mlsanity.com/paper/2301.06806/) (2023)
- [The Ball-Proximal (="Broximal") Point Method: a New Algorithm, Convergence Theory, and Applications](https://mlsanity.com/paper/2502.02002/) (2025)
- [Stochastic Smoothed Primal-Dual Algorithms for Nonconvex Optimization with Linear Inequality Constraints](https://mlsanity.com/paper/2504.07607/) (2025)
- [On the Convergence of FedProx with Extrapolation and Inexact Prox](https://mlsanity.com/paper/2410.01410/) (2024)
- [LEAF: A Learning-Enabled ADMM Framework for Accelerated Convex Optimization](https://mlsanity.com/paper/2606.08993/) (2026)

## Abstract (arXiv)

In this work, using Moreau envelopes, we define a complete metric for the set of proper lower semicontinuous convex functions. Under this metric, the convergence of each sequence of convex functions is epi-convergence. We show that the set of strongly convex functions is dense but it is only of the first category. On the other hand, it is shown that the set of convex functions with strong minima is of the second category.