Convex functions on compact C^*-convex sets

It is well known that if a real valued convex function on a compact convex domaincontained in the real numbers attains its maximum,then it does so at least at one extreme point of its domain.In this paper,we consider a matrix convex function on a compact and C^*-convex set generated by self--adjoint matrices.An important issue is so that this function on a compact and C^*-convex domain attains its maximum at a C^*-extreme point.