A generalization of the Chandler Davis convexity theorem

In 1957 Chandler Davis proved a theorem that a rotationally invariant function on symmetric matrices is convex if and only if it is convex on the diagonal matrices. We generalize this result to groups acting nonlinearly on convex subsets of arbitrary vector spaces thereby understanding the abstract mechanism behind the classical theorem. We apply the new theorem to a problem from the mathematical theory of composite materials and derive its corollaries in the Lie algebra setting. Using the latter, we show that the Pfaffian is log-concave.