On simplexes inscribed in a hypersurface

M. L. Gromov

doi:10.1007/BF01098717

On simplexes inscribed in a hypersurface

M. L. Gromov

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	52-56
Number of pages	5
Journal	Mathematical Notes of the Academy of Sciences of the USSR
Volume	5
Issue number	1
DOIs	https://doi.org/10.1007/BF01098717
State	Published - Jan 1969

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General Mathematics

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title = "On simplexes inscribed in a hypersurface",

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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AB - 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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On simplexes inscribed in a hypersurface

Abstract

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