Convex functions on compact $C^*$-convex sets

Document Type : Research Paper


Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Islamic Republic of Iran.



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.


[1] J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. Am. Math. Soc., 79 (1955), 58-71.
[2] J.-C. Bourin, Hermitian operators and convex functions, JIPAM, J. Inequal. Pure Appl. Math., 6(5) (2005), Art 139.
[3] J.-C. Bourin and E.-Y. Lee, Unitary orbits of Hermitian operators with convex or concave functions, Bull. Lond. Math. Soc., 44 (2012), 1085-1102.
[4] D.R. Farenick, $C^*$-convexity and matricial ranges, Can. J. Math., 44 (1992), 280-297.
[5] D.R. Farenick, Krein-Milman type problems for compact matricially convex sets, Linear Algebra Appl., 162-164 (1992), 325-334.
[6] D.R. Farenick and P.B. Morenz, $C^*$-extreme points of some compact $C^*$-convex sets, Proc. Am. Math. Soc., 118 (1993), 765-775.
[7] K. L$rmddot{o}$wner, $ddot{U}$ber monotone Matrix funktionen, Math. Z., 38 (1934), 177-216.
[8] B. Magajna, $C^*$-convexity and the Numerical Range, Can. J. Math., 43(2) (2000), 193-–207.
[9] B. Magajna, On $C^*$-extreme points, Proc. Am. Math., Soc., 129(3) (2000), 771-780.
[10] P.B. Morenz, The structure of $C^*$-convex sets, Can. J. Math., 46(5) (1994), 1007-1026.