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

TL;DR

This paper refines Minkowski's 3^d-theorem by establishing an upper bound on the number of lattice directions with minimal width in convex bodies, with equality only for the regular cross-polytope, supported by two independent proofs.

Contribution

It provides a sharpened version of Minkowski's 3^d-theorem and characterizes cases of equality, offering new insights into lattice width directions in convex geometry.

Findings

Maximum of 3^d-1 lattice directions with minimal width

Equality holds only for the regular cross-polytope

Two independent proofs of the sharpened 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.