A note on the K-epigraph

Abstract

We study the question as to when the closed convex hull of the graph 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.

Keywords:

• K-convexity
• K-epigraph
• horizon cone
• spectral function
• matrix analysis

MSC 2000 Subject Classifications:

• 52A41
• 65K10
• 90C25
• 90C46

Acknowledgments

The authors would like to thank two anonymous referees for their extremely valuable comments They would also like to thank Prof. James V. Burke (University of Washington) for very useful discussions on the material presented here.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here $\lambda \left(X\right)=({\lambda }_1,\dots ,{\lambda }_n{)}^T$ is the vector of eigenvalues of $X\in {\mathbb{S}}^n$ in decreasing order.

2 We refer the reader to Definition 4.9 for a formal introduction.

Additional information

Funding

This work was supported by Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [RGPIN-2017-04035].