Lattice-Width Directions and Minkowski's $3^d$-Theorem
Jan Draisma, Tyrrell B. McAllister, Benjamin Nill · SIAM Journal on Discrete Mathematics · 2012
We show that the number of lattice directions in which a convex body in $\mathbb{R}^d$ 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 Minkowski.