Two results on the Convex Algebraic Geometry of sets with continuous symmetries

TL;DR

This paper explores the convex algebraic geometry of sets with continuous symmetries, proving that invariant spectrahedra can be described equivariantly and analyzing the preservation of convex set classes under Kostant's theorem.

Contribution

It establishes that invariant spectrahedra admit equivariant descriptions and shows the preservation of convex set classes under a key bijection in symmetric spaces.

Findings

Invariant spectrahedra have equivariant linear matrix inequality descriptions.

The bijection preserves convex semi-algebraic sets, spectrahedral shadows, and rigidly convex sets.

Results connect convex geometry with symmetry groups in Euclidean spaces.

Abstract

We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, i.e., can be described by an equivariant linear matrix inequality. Second, we show that the bijection induced by Kostant's Convexity Theorem between convex subsets invariant under a polar representation and convex subsets of a section invariant under the Weyl group preserves the classes of convex semi-algebraic sets, spectrahedral shadows, and rigidly convex sets.