A Geometric Take on Kostant’s Convexity Theorem

Given a compact Lie group \(G\) $G$ and an orthogonal \(G\) $G$ -representation \(V\) $V$ , we give a purely metric criterion for a closed subset of the orbit space \(V/G\) $V/G$ to have convex pre-image in \(V\) $V$ . In fact, this also holds with the natural quotient map \(V\rightarrow V/G\) $V\to V/G$ replaced with an arbitrary submetry \(V\rightarrow X\) $V\to X$ . In this context, we introduce a notion of “fat section” which generalizes polar representations, representations of non-trivial copolarity, and isoparametric foliations. We show that Kostant’s Convexity Theorem partially generalizes from polar representations to submetries with a fat section, and give examples illustrating that it does not fully generalize to this situation.