# When is the log-permanent concave?

Let $$\operatorname{PSD}_n$$ be the cone of $$n\times n$$ semidefinite positive matrices. For any $$X\in \operatorname{PSD}_n$$, define $$f(X)=\log(\det(X)).$$ Then $$f$$ is a concave function on $$\operatorname{PSD}_n$$. This fact has some significance in convex optimization.

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

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

EDIT: In the comments below, Mark L. Stone included a reference to a very interesting theorem; here's the relevant part.

Let $$H_n$$ be the set of $$n\times n$$ Hermitian matrices. For $$X,Y\in H_n$$, we say that $$X\preccurlyeq Y$$ iff $$Y-X$$ is semidefinite positive. Let $$S_n$$ be the group of permutations on $$n$$ objects, and $$G$$ a subgroup of $$S_n$$.

We define a function $$d:H_n \rightarrow R$$ for an irreducible character $$\chi$$ of $$G$$ by: $$d(X)=\sum_{\sigma\in G} \chi(\sigma) \prod_{i=1}^{n} X_{\sigma(i),i}$$

(Note that the determinant and the permanent are instances of this function, as are, I think, the immanants.)

Theorem: If $$X,Y\in H_n$$, $$0 \preccurlyeq Y \preccurlyeq X$$ and $$0\leq \lambda \leq 1$$, then $$d$$ is convex, i.e., $$d(\lambda X + (1-\lambda)Y) \leq \lambda d(X) + (1-\lambda)d(Y)$$

In particular, that holds for the permanent. However, this result doesn't quite answer my original question, in that there is no fixed set $$M$$ that works— it requires a relationship between $$X$$ and $$Y$$. (Is there some way to lift this result with a matrix exponential to get $$M=\operatorname{PSD}_n$$?)

Consider the set of matrices with positive elements. The permanent (or whenever the "character" of the immanent) is a posynomial (polynomial with positive coefficients). Making a change of variables for each element as $$x=e^y$$ and taking log of the permanent, one gets it in the form of a log-sum-exp function. This function is convex in $$y$$.