Lattice inequalities for convex bodies and arbitrary lattices
For a convex body \(K\) in Euclidean \(E^ d\) let \(V(K)\) and \(F(K)\) denotes its volume and surface area. If \(L \subset E^ d\) is a lattice (with \(\text{det} L>0)\), then \(G(K,L)=\text{card} (K \cap L)\) denotes the lattice point enumerator. Further \(\lambda_ d(L)\) denotes the last of the classical successive minima of \(L\) with respect to the unit ball \(B^ d\). Then the author proves \[ G(K,L) \text{det} L \geq V(K)-{1 \over 2} \lambda_ d(L) F(K) \] which is tight for orthogonal lattice and generalizes a result by \textit{J. Bokowski}, \textit{H. Hadwiger} and the reviewer [Math. Z. 127, 363-364 (1972; Zbl 0238.52005)]. The author further proves for ``lattice-periodic'' sets: \[ \lambda_ d(L) F(A) \geq 8V(A)(1-V(A)/ \text{det} L) \] again generalizing a classical isoperimetric inequality by \textit{H. Hadwiger} [Monatsh. Math. 76, 410-418 (1972; Zbl 0248.52012)].
scientific article; zbMATH DE number 508791