Topological properties of extreme points of convex compact sets in $\ell^2$

Under certain restrictions on given sets $M$ and $K$, where $M\subset K$ and $K$ is a metric compact set, a continuous map $\varepsilon\colon K\to\ell^2$ is constructed such that $\operatorname{ext}\operatorname{conv}\varepsilon(K)=\varepsilon(M)$ and the restriction of $\varepsilon$ to $M$ is a topological embedding. Here $\operatorname{ext}$ is the set of extreme points and $\operatorname{conv}$ is the closed convex hull.