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I apologize if the following appears incoherent or incorrect; I am trying to cobble together ideas from various sources (don't ask me to provide the sources I am using--far too many, and I can't find them all). Anyways, suppose I know that both $A$ and $B$ are convex sets in $M_n (\Bbb{C})$, each generated from their collection of extreme points. What can I say about the extreme points of $A \times B$? How are they related to the extreme points of each individual factor? If we need additional hypothesis on $A$ and $B$ in order to have a relatively substantial question, please suggest them.

Another question I have is about the extreme points of a convex subset of a convex set. Specifically, if $C$ is convex set in $M_n(\Bbb{C})$, and $S$ is a convex subset of $C$, is there any possible relationship between their extreme points? Again, if additional hypothesis are need, please feel free to suggest them.

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It's straightforward to show that $(a, b)$ is extreme in $A \times B$ if and only if $a$ and $b$ are extreme in $A$ and $B$ respectively.

As for your second question, in general, the extreme points of a convex subset of a convex set need not bear any special relationship to the extreme points of the superset. If you can show that $B$ is a face of $A$ (that is, whenever $b = \lambda a_1 + (1 - \lambda) a_2$ for $a_1, a_2 \in A$, $b \in B$, and $\lambda \in (0, 1)$, we have $a_1, a_2 \in B$), then it is true that the extreme points of $B$ are a subset of the extreme points of $A$. Other than that, I don't know of any other conditions sufficient for special relationships between the extreme points of nested convex sets.

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  • $\begingroup$ There is Dubins's theorem: if $E$ is a real linear space, $K$ a convex subset of $E$ which is linearly closed and linearly bounded, and $M$ a flat of finite codimension $n$ in $E,$ then each extreme point of $K\cap M$ is a convex combination of at most $n+1$ extreme points of $K$. There is a paper by Victor Klee "On a Theorem of Dubins" and one by Dubins, "On extreme points of convex sets", both in J. Math. Anal. Appl., in 1963 and 1962 respectively. $\endgroup$ – kimchi lover Jul 24 '17 at 0:43

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