Distance to the Convex Hull of an Orbit under the Action of a Compact Lie Group

For a real vector space V acted on by a group K and fixed x and y in V , we consider the problem of finding the minimum (resp., maximum) distance, relative to a Kinvariant convex function on V , between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V . Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component p of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.