The Optimal Smoothings of Sublinear Functions and Convex Cones

TL;DR

This paper characterizes the set of optimal smoothings for convex functions and cones, providing a comprehensive framework for approximating non-smooth convex objects with smooth ones, with applications to sublinear functions and convex sets.

Contribution

It offers a full characterization of all optimal smoothings for convex cones and sublinear functions, including inner and outer approximations, and applies this to complex convex structures.

Findings

Complete characterization of optimal smoothings for convex cones and sublinear functions

Provides conditions for inner and outer smooth approximations

Applies theory to compositions with sublinear functions and convex sets

Abstract

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.