The talk concerns inequalities on functions of matrix variables, in particular a convex set C of matrices defined by them. The functions are typically (noncommutative) polynomials or rational functions and the sets include matrices of all sizes, hence are dimension free convex sets. Extreme points are getting to be understood. But optimizing a linear functional leads in our experiments to surprising properties which are unexplained. The talk describes the sets, the extreme points and the experimental findings.