Let's begin with a few observations. Suppose we consider the set of $N\times N$ matrices and consider the matrices with positive determinant. There are several connected components in this set; let $S$ be the component that contains the identity matrix. Observe that $S$ corresponds to the set of positive definite matrices. This fact (and the convexity of $S$) is important for interior-point techniques in semi-definite programming.

Suppose we replace "determinant" with "permanent". What happens? In other words, suppose we consider matrices with positive permanent. Let $C$ be the connected component that contains the identity matrix. The closure $\overline{C}$ is a pointed cone.

- Is $C$ convex?
- Does $C$ or $\overline{C}$ have a name, or can anyone point me to literature discussing its properties?