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The isodiametric problem with lattice-point constraints

by M.A. Hernandez Cifre, A. SchÜrmann, F. Vallentin.

Series: 2007-45, Preprints

52A20 Convex sets in $n$ dimensions (including convex hypersurfaces)
52C07 Lattices and convex bodies in $n$ dimensions
52A40 Inequalities and extremum problems

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.

Isodiametric problem; lattices; Dirichlet-Voronoi cells parallelohedra

This paper was published in:
Monatshefte für Mathematik