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With reference to this paper.

Let $V$ be a locally convex space, and $K=(K_n)$ be a compact matrix convex set in $V$. Then as proved in Cor 3.6 in the above paper, we see that if $v\in K_n$ is a matrix extreme point, then it is also extreme in the usual sense. What about the converse? Of course I guess that not every extreme point is a matrix extreme point and I think the answer lies in the example of matrix intervals $[aI,bI]$. But I am not able to produce a specific example.

Am I looking in the right place? Also, is there some criteria when both sets coincide?

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