at the moment I am preparing for an exam in optimization and my tutor mentioned something regarding extreme points of convex sets (If a set $X$ is convex set then $x\in X $ is extreme iff $\forall y,z\in X,\forall\lambda\in [0,1]:\lambda y+(1-\lambda)z\neq x)$. He said that there is a way of showing that a point $x$ is extreme by finding a convex function with minimum at $x$ and then doing something with Brouwer's fixed point theorem. I honestly did not really understand it. Is anybody able to explain hoe exactly this works?
Proving that a point is an extreme point of convex set using Brouwer Fixed-Point theorem
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convex-analysis
convex-optimization