Does anyone know the answer (and proof) to this question: Suppose $S\subset {\mathbb R}^{m\times n}$ is a convex and compact subset of $m\times n$ real matrices with respect to the Frobenius norm, and suppose for each $v\in {\mathbb R}^m$ the set $\{v^Ts:s\in S\}$, contains the zero vector in ${\mathbb R}^n$. Is it true that $S$ contains the zero matrix?
When $n=1$, this is of course a well known and simple result.