Extension of Convex Function

TL;DR

This paper investigates the extension of convex functions from non-convex domains to their convex hulls, analyzing both local and global convexity, including cases defined by positive definite Hessians.

Contribution

It provides new theoretical results on extending convex functions and convexity defined by positive definite Hessians beyond their original domains.

Findings

Characterization of conditions for extending convex functions

Extension results for convex functions with positive definite Hessians

Parallel results for local and global convexity

Abstract

We study the local and global versions of the convexity, which is closely related to the problem of extending a convex function on a non-convex domain to a convex function on the convex hull of the domain and beyond the convex hull. We also give the parallel results for the convexity defined by positive definite Hessian.