Let $S$ be the set of $n\times n$ semidefinite positive matrix. For any $X\in S$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function over $S$. This fact has some significance in convex optimization.

Is there an analogous result for the permanent? In particular, if we define $$g(X)=\log(Perm(X)),$$ can we identify some non-trivial set $M\subseteq R^{n\times n}$ of matrices over which $g$ is concave (or convex for that matter)?

(I'm hoping for some set larger than "diagonal matrices".)