  Let G be a complex semisimple Lie group, K a maximal compact subgroup and V
an irreducible representation of K. Denote by M the unique closed orbit of G in
P(V) and by O its image via the moment map. For any measure on M we construct a
map from the Satake compactification of G/K (associated to V) to the Lie
algebra of K. For the K-invariant measure, this map is a homeomorphism of the
Satake compactification onto the convex envelope of O. For a large class of
measures the image of the map is the convex envelope. As an application we get
sharp upper bounds for the first eigenvalue of the Laplacian on functions for
an arbitrary Kaehler metric on a Hermitian symmetric space.

Differential Geometry;Representation Theory