On simplexes inscribed in a hypersurface

The sufficient conditions are obtained for the existence, on a hyper surface $M\subset R^n$, of $k$ points whose convex hull forms a $(k-1)$-dimensional simplex, homothetic to a given simplex $\Delta\subset R^n$. In particular, it is shown that if $M$ is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex $\Delta\subset R^n$. The proofs are based on simple topological considerations. There are six references.