Pome07EC0797039CProposition 11.29.4.07EC0797039C At each fixed $m$, exactly one of the following holds:
Some Hermitian matrix $B$ satisfies $\operatorname {tr}(B\Psi _ m)>0$ everywhere on $K$.
There is a probability measure $\mu $ on $K$, supported on at most $N_ m^2+1$ points, such that $\int _ K\Psi _ m\, \mathrm d\mu =0$.
The first alternative holds precisely when $0\notin \operatorname {conv}(\Psi _ m(K))$.
Proof. The real vector space of Hermitian $N_ m\times N_ m$ matrices has dimension $N_ m^2$. Carathéodory's theorem expresses every point of the convex hull as a convex combination of at most $N_ m^2+1$ points of the compact image $\Psi _ m(K)$, so the hull is compact. If it contains zero, the coefficients of such an expression define the required atomic probability measure $\mu $, and then $\int _ K\operatorname {tr}(B\Psi _ m)\, \mathrm d\mu =0$ excludes the first alternative.
If the hull misses zero, let $Z_0$ be its point of least Hilbert–Schmidt norm. Differentiating the squared norm along the segment from $Z_0$ to any $Z$ in the hull gives
Taking $B=Z_0$ proves the first alternative. $\square$
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