# Proximal mappings and Moreau envelopes of single-variable convex piecewise cubic functions and multivariable gauge functions

arXiv: https://arxiv.org/abs/1909.05795 · HTML: https://mlsanity.com/paper/1909.05795/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2019.

## Built on by (most cited first, 4 of 4)

- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [A Non-Asymptotic Moreau Envelope Theory for High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/2210.12082/) (2022)
- [The Ball-Proximal (="Broximal") Point Method: a New Algorithm, Convergence Theory, and Applications](https://mlsanity.com/paper/2502.02002/) (2025)
- [On the Convergence of FedProx with Extrapolation and Inexact Prox](https://mlsanity.com/paper/2410.01410/) (2024)

## Abstract (arXiv)

This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their Moreau envelopes categorized. Characterizations of convex Moreau envelopes are established; topics include strict convexity, strong convexity and Lipschitz continuity.