Convexity of the gradient map image μ_p(X) in the general compact smooth case

Determine whether the image μ_p(X) ⊂ 𝔭 of the restricted momentum (gradient) map μ_p: X → 𝔭 for a real reductive subgroup G ⊂ U^ compatible with the Cartan decomposition is convex when X is a compact smooth G-invariant submanifold of Z; current results only show that μ_p(X) is a finite union of convex polytopes.

Background

The paper studies the image of the restricted momentum (gradient) map μ𝔭: X → 𝔭, whose convex hull E and its extreme points underpin the structure theorem proved by the authors. In the complex connected case, μ𝔭(X) is known to be convex (Atiyah–Guillemin–Sternberg).

For general compact smooth X under a real reductive G, only a weaker result is known: μ𝔭(X) is a finite union of convex polytopes. The authors explicitly state that the convexity of μ𝔭(X) in this general setting remains an open question (see also Heinzner–Schützdeller). Resolving this would sharpen understanding of the geometry of the momentum/gradient map image central to the paper’s results.