  Given a compact Lie group $G$ and an orthogonal $G$-representation $V$, we
give a purely metric criterion for a closed subset of the orbit space $V/G$ to
have convex pre-image in $V$. In fact, this also holds with the natural
quotient map $V\to V/G$ replaced with an arbitrary submetry $V\to X$.
  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.
