Around Minkowski's 3d - 1 Theorem

Jan Draisma

Today, one hundred years and two days after Hermann Minkowski died, I will discuss a recent result intimately related with a famous theorem by Minkowski alluded to in the title.

This result, by Benjamin Nill, Tyrrell McAllister, and myself, is a d-dimensional generalisation of the following proposition: the diamond with vertices (1,0), (0,1), (-1,0), (0,-1) is, up to scaling and translating and integral linear transformations, the unique closed, convex subset S of R2 whose "lattice width" is attained in the maximal possible number of lattice directions, namely, 8. Here "lattice width" is the infimum of (sup v(S) - inf v(S)) over all v in the lattice dual to Z2 < R2.

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