Convex functions on compact C∗-convex sets

It is well known that if a real valued convex function on a compact convex domain contained 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.