In representations with nontrivial copolarity, faces of G-invariant compact convex sets are determined by their intersections with fat sections, including exposure of faces.
Magata, A general Weyl-type integration formula for isometric group actions, Transformation Groups,15(1), 184–200 (2010)
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Faces of invariant convex sets in representations of nontrivial copolarity
In representations with nontrivial copolarity, faces of G-invariant compact convex sets are determined by their intersections with fat sections, including exposure of faces.