On the Distance of Polytopes with Few Vertices to the Euclidean Ball

Konstantin E. Tikhomirov

doi:10.1007/s00454-014-9639-9

On the Distance of Polytopes with Few Vertices to the Euclidean Ball

Konstantin E. Tikhomirov

Mathematics

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

2 Scopus citations

Abstract

Let n, N be natural numbers satisfying (Formula presented.) be the unit Euclidean ball in Rⁿ, and let (Formula presented.) be a convex n-dimensional polytope with N vertices and the origin in its interior. We prove that (Formula presented.) where c>0 is a universal constant. As an immediate corollary, for any covering of S^n-1 by N spherical caps of geodesic radius ϕ, we get that (Formula presented.) for an absolute constant C>0. Both estimates are optimal up to the constant multiples c, C.

Original language	English (US)
Pages (from-to)	173-181
Number of pages	9
Journal	Discrete and Computational Geometry
Volume	53
Issue number	1
DOIs	https://doi.org/10.1007/s00454-014-9639-9
State	Published - Jan 2015

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Keywords

Banach–Mazur distance
Convex polytope
Covering by spherical balls

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abstract = "Let n, N be natural numbers satisfying (Formula presented.) be the unit Euclidean ball in Rn, and let (Formula presented.) be a convex n-dimensional polytope with N vertices and the origin in its interior. We prove that (Formula presented.) where c>0 is a universal constant. As an immediate corollary, for any covering of Sn-1 by N spherical caps of geodesic radius ϕ, we get that (Formula presented.) for an absolute constant C>0. Both estimates are optimal up to the constant multiples c, C.",

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AU - Tikhomirov, Konstantin E.

PY - 2015/1

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N2 - Let n, N be natural numbers satisfying (Formula presented.) be the unit Euclidean ball in Rn, and let (Formula presented.) be a convex n-dimensional polytope with N vertices and the origin in its interior. We prove that (Formula presented.) where c>0 is a universal constant. As an immediate corollary, for any covering of Sn-1 by N spherical caps of geodesic radius ϕ, we get that (Formula presented.) for an absolute constant C>0. Both estimates are optimal up to the constant multiples c, C.

AB - Let n, N be natural numbers satisfying (Formula presented.) be the unit Euclidean ball in Rn, and let (Formula presented.) be a convex n-dimensional polytope with N vertices and the origin in its interior. We prove that (Formula presented.) where c>0 is a universal constant. As an immediate corollary, for any covering of Sn-1 by N spherical caps of geodesic radius ϕ, we get that (Formula presented.) for an absolute constant C>0. Both estimates are optimal up to the constant multiples c, C.

KW - Banach–Mazur distance

KW - Convex polytope

KW - Covering by spherical balls

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JO - Discrete and Computational Geometry

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ER -

On the Distance of Polytopes with Few Vertices to the Euclidean Ball

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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