When is the inverse of an invertible convex function itself convex?

TL;DR

This paper establishes a sufficient condition under which the inverse of an invertible convex function remains convex, focusing on the properties of the gradient of the inverse's components.

Contribution

It introduces a novel sufficient condition involving the sign of the inverse's gradient components to ensure the convexity of the inverse function.

Findings

Inverse convexity is preserved when the gradient of each inverse component has negative entries.

The paper provides conditions for local strong convexity of the inverse.

Results apply to vector-valued functions on bR^N.

Abstract

We provide a sufficient condition for an invertible (locally strongly) convex vector-valued function on $\mathbb{R}^N$ to have a (locally strongly) convex inverse. We show under suitable conditions that if the gradient of each component of the inverse has negative entries, then this inverse is (locally strongly) convex if the original is.