Strict linear alternative for GCM trichotomy

If every $v_i\cdot x>0$, then $0=({\sum }_i{t}_i{v}_i)\cdot x={\sum }_i{t}_i(v_i\cdot x)$ with $t_i\ge 0$ forces each $t_i=0$. If $m=0$, take $x=0$ and both conditions hold vacuously. If $n=0<m$, every row is zero, so neither condition holds. Thus the reverse direction need only consider $m,n\ge 1$.

Assume there is no nonzero nonnegative relation. The coefficient simplex $E=\{t\in {\mathbb{R}}^m:t_i\ge 0,{\sum }_i{t}_i=1\}$ is nonempty, closed and bounded. The polynomial function $t\mapsto \mathrm{\parallel }{\sum }_i{t}_i{v}_i{\mathrm{\parallel }}^2$ is continuous, so F1 supplies a minimizer $t^{\ast }$. Put $x={\sum }_i{t}_i^{\ast }{v}_i$. This vector is nonzero by the hypothesis. For each convex combination $y={\sum }_i{s}_i{v}_i$, the coefficients $(1-t)t^{\ast }+ts$ remain in $E$ for $0\le t\le 1$. Minimality gives $0\le 2t(x,y-x)+t^2\mathrm{\parallel }y-x{\mathrm{\parallel }}^2$. For $t>0$, divide by $t$; if $(x,y-x)<0$, sufficiently small positive $t$ contradicts the inequality. Therefore $x\cdot y\ge \mathrm{\parallel }x{\mathrm{\parallel }}^2>0$, in particular $x\cdot v_i>0$. This proves the reverse implication without a separate compact-image assumption.

Apply the equivalence to the rows of $-C$ together with the coordinate rows of the identity matrix. A nonnegative relation has the form $-C^t\lambda +\mu =0$ with $\lambda \ge 0$, $\mu \ge 0$. The matrix hypothesis forces $\lambda =0$, and hence $\mu =0$. The separating vector $v$ thus satisfies $-Cv>0$ and $v>0$. If either matrix dimension is zero the same empty-coordinate interpretation applies; for $n=0<m$ the matrix hypothesis is false.