Lattice-width directions and Minkowski's 3^d-theorem

J. Draisma; T.B. McAllister; B. Nill

doi:10.1137/120877635

Lattice-width directions and Minkowski's 3^d-theorem

J. Draisma, T.B. McAllister, B. Nill

Discrete Algebra and Geometry

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

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Abstract

We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864--12 January 1909), for which we provide two independent proofs.

Original language	English
Pages (from-to)	1104-1107
Journal	SIAM Journal on Discrete Mathematics
Volume	26
Issue number	3
DOIs	https://doi.org/10.1137/120877635
Publication status	Published - 2012

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title = "Lattice-width directions and Minkowski's 3^d-theorem",

abstract = "We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864--12 January 1909), for which we provide two independent proofs.",

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AB - We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864--12 January 1909), for which we provide two independent proofs.

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JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

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Lattice-width directions and Minkowski's 3^d-theorem

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