Isoperimetric inequalities for densities of lattice-periodic sets
Lattice-periodic sets have infinite volume and surface area. These can be replaced by appropriate finite measures, the volume density and the surface area density, both measured with respect to the fundamental cell. For these densities there exist isoperimetric inequalities, analogously to the classical ones. The first one was discovered by H. Hadwiger in 1972 for the standard lattice \(\mathbb{Z}^d\), and he determined also the extremal cases. In 1993 U. Schnell found the corresponding isoperimetric inequality for general lattices in terms of Minkowski's last successive minimum. Here the author gives another interesting isoperimetric inequality in terms of minimal determinants of sublattices; proved for \(d=2\) and conjectures for \(d\geq 3\).
scientific article; zbMATH DE number 1300154