The isodiametric problem with lattice-point constraints

TL;DR

This paper investigates the isodiametric problem for symmetric convex bodies in Euclidean space constrained by lattice points, identifying extremal shapes and proving conjectures in specific cases.

Contribution

It introduces a new extremal body characterized by intersections with Dirichlet-Voronoi cells and proves the conjecture for three-dimensional and select lattices.

Findings

Intersection with Dirichlet-Voronoi cell is extremal.

Conjecture proven for 3D and certain lattices.

Identifies minimal diameter bodies with fixed volume.

Abstract

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.