On homeomorphisms of hyperspaces of convex subsets

The famous Shchepin's results on spectral analisys of homeomorphisms of normal functor-powers in tne category of compacta are exstended to "normal" functors from the category of convex compacta to the category of compacta. It is stated that for a metrizable convex compacta $K$, $L$, $M$, which are distinct from a point, and $\tau\ge\omega_2$ the spaces $K^\tau$, the hyperspace of convex closed subsets of $L^\tau$ and the hyperspace of convex closed subsets of $M^\tau$, which are convex hulls of $\le n$ points, $n\ge2$, are mutually non-homeomorphic. For any homeomorphism between the hyperspaces of convex closed subsets of $K^\tau$ and $L^\tau$ the preservation of finite-dimensional elements and finite-dimensional elements with $n$ extreme points for $n\in A$, $A$ necessarily contains $1$ and $2$ and coincides with $\Bbb N$ or its initial segment, is proved.