Faces of invariant convex sets in representations of nontrivial copolarity

TL;DR

This paper investigates the face structure of invariant convex sets in orthogonal representations of compact Lie groups with nontrivial copolarity, generalizing previous results for polar representations.

Contribution

It establishes that the face structure of such convex sets is fully determined by their intersection with a fat section, extending known results to nonpolar cases.

Findings

Face structure of invariant convex sets is determined by intersection with fat section.

A face is exposed if and only if the corresponding face in the section is exposed.

Generalizes previous polar representation results to nonpolar cases.

Abstract

Let $(V, G)$ be an orthogonal representation of a compact Lie group $G$ with nontrivial copolarity, and $\Sigma$ a fat section of $(V, G)$. If $E$ is a $G$-invariant compact convex set in $V$, then $P=E\cap\Sigma$ is a convex set in $\Sigma$. We prove that up to conjugacy the face structure of $E$ is completely determined by that of $P$ and that a face of $E$ is exposed if and only if the corresponding face of $P$ is exposed. Our result generalizes the result proved by Leonardo Biliotti, Alessandro Ghigi and Peter Heinzner in the case where $(V, G)$ is a polar representation.