# The Optimal Smoothings of Sublinear Functions and Convex Cones

arXiv: https://arxiv.org/abs/2508.06681 · HTML: https://mlsanity.com/paper/2508.06681/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Aug 2025.

## Built on by (most cited first, 1 of 1)

- [An Elementary Proof of the Near Optimality of LogSumExp Smoothing](https://mlsanity.com/paper/2512.10825/) (2025)

## Abstract (arXiv)

This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for any target level of smoothness. Optimal smoothings restricting to either inner or outer approximations also follow from our theory. Finally, we apply our theory to provide insights into smoothing amenable functions given by compositions with sublinear functions and generic convex sets by expressing them as conic sections.