Abstract
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 Δ ⊂ Rn. In particular, it is shown that if M is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex Δ ⊂ Rn. The proofs are based on simple topological considerations. There are six references.
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Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 81–89, January, 1969.
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Gromov, M.L. On simplexes inscribed in a hypersurface. Mathematical Notes of the Academy of Sciences of the USSR 5, 52–56 (1969). https://doi.org/10.1007/BF01098717
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DOI: https://doi.org/10.1007/BF01098717