On simplexes inscribed in a hypersurface

The sufficient conditions are obtained for the existence, on a hyper surface M Rⁿ, of k points whose convex hull forms a (k–1)-dimensional simplex, homothetic to a given simplex Rⁿ. In particular, it is shown that if M is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex Rⁿ. The proofs are based on simple topological considerations. There are six references.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 81–89, January, 1969.