I am actually trying to solving the following problem from Matrix analysis by Fumio Hiai and Denes Petz.
Show that the extreme points of the set $$ \mathcal{S}_{n}:=\left\{D \in \mathbb{M}_{n}^{s a}: D \geq 0 \text { and } \operatorname{Tr} D=1\right\} $$ are the orthogonal projections of trace 1 . Show that for $n>2$ not all points in the boundary are extreme.
I was thinking about whether I could find a point on the boundary. Then, I could show that it could be decomposed by a linear combination of extreme points.
The convex set I am more familiar with is the convex polytopes. But the idea of a point on the boundary is not clear to me here.
For n=2, there is a parametrization of the matrices,
$$ \frac{1}{2}\left[\begin{array}{cc} 1+\lambda_{3} & \lambda_{1}-\mathrm{i} \lambda_{2} \\ \lambda_{1}+\mathrm{i} \lambda_{2} & 1-\lambda_{3} \end{array}\right]=\frac{1}{2}\left(I+\lambda_{1} \sigma_{1}+\lambda_{2} \sigma_{2}+\lambda_{3} \sigma_{3}\right) $$ where $\sigma_{1}, \sigma_{2}, \sigma_{3}$ are the Pauli matrices and the necessary and sufficient condition to be in $\mathcal{S}_{2}$ is $$ \lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2} \leq 1. $$ How do I say a matrix is on the boundary of $\mathcal{S}_{2}$?
Does it mean $(\lambda_1, \lambda_2, \lambda_3)$ need to fit in certain conditions or compare their norm or something?