The isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space 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. Authors’ addresses: M. A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain; A. Schürmann, Institut für Algebra und Geometrie, Otto-von-Guericke Universit?t Magdeburg, 39106 Magdeburg, Germany; F. Vallentin, Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands