I have a real symmetric strictly positive definite $n\times n$ matrix $V$. Let $C$ be the positive cone $\{x\ge0\}$ in $\mathbb{R}^n$. Is the set $S=\{\>x\in C\>|\>Vx\in C\>\}$ non-empty (except for $x=0$)? In fact, does $S$ have a nonempty interior? Another way to say this is, does the image $V^{-1}C$ intersect $C$ itself in a nontrivial way?
I have been thinking a lot about the geometry at least in $\mathbb{R}^3$ and it seems that this should be true. But it seems that this should follow directly from the nature of symmetric positive definite matrices.
Many thanks for any help.