INSCRIBING A REGULAR OCTAHEDRON IN A THREE-DIMENSIONAL CONVEX BODY WITH SMOOTH BOUNDARY


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Abstract

A norm ∥ · ∥ and a convex body K with smooth boundary in the standard Euclidean space ℛR3 are considered. It is proved that the boundary ∂K of K contains the vertices AA′BB′CC′ of a regular octahedron with ∥AA′∥ = ∥BB′∥ ≥ ∥CC′∥ (or, optionally, ∥AA′∥ = ∥BB′∥ ≤ ∥CC′∥). Bibliography: 4 titles.

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