A note on the K-epigraph

TL;DR

This paper investigates conditions under which the closed convex hull of a K-convex map coincides with its K-epigraph, exploring the minimal cone K for convexity and closedness, with applications in matrix space and convex functions.

Contribution

It introduces criteria for the equality of convex hulls and K-epigraphs, and characterizes the smallest cone K ensuring convexity and closedness of the K-epigraph.

Findings

Identifies conditions for convex hull and K-epigraph equality.

Characterizes the minimal cone K for convexity.

Provides applications in matrix space and convex functions.

Abstract

We study the question as to when the closed convex hull of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions.