Inscribed and circumscribed polyhedra for a convex body and continuous functions on a sphere in Euclidean space

Makeev, V.

doi:10.1090/S1061-0022-07-00979-X

Inscribed and circumscribed polyhedra for a convex body and continuous functions on a sphere in Euclidean space
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by V. V. Makeev
Translated by: B. M. Bekker

St. Petersburg Math. J. 18 (2007), 997-1009

DOI: https://doi.org/10.1090/S1061-0022-07-00979-X

Published electronically: October 2, 2007

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Abstract:

Two related problems concerning continuous functions on a sphere $S^{n-1}\subset {\mathbb {R}}^n$ are studied, together with the problem of finding a family of polyhedra in ${\mathbb {R}}^n$ one of which is inscribed in (respectively, circumscribed about) a given smooth convex body in ${\mathbb {R}}^n$. In particular, it is proved that, in every convex body $K\subset {\mathbb {R}}^3$, one can inscribe an eight-vertex polyhedron obtained by “equiaugmentation” of a similarity image of any given tetrahedron of class $T$.

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