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".)