The isodiametric problem with lattice-point constraints (Q956673)

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean \(d\)-space \({\mathbb R}^d\) 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. The main result states: Let \(L\) be~a full rank lattice in~\({\mathbb R}^d\) and let \(V\in (0, 2^d\det L]\) be a~real number. Suppose that \(K\) is a~centrally symmetric convex body in~\({\mathbb R}^n\) with \(\text{vol}\,K = V\). Then \(\text{diam} K_L(V)\leqslant \text{diam}K\). Here, \(\det L\) is the determinant of~\(L\) and \(K_L(V) = B_d(r_L(V))\cap \text{DL}(2L)\), where \(\text{DL}(L)\) is the Dirichlet-Voronoi cell of~\(L\), and \(r_L(V)\) is the (unique) real number such that \(K_L(V)\) has volume~\(V\). It is also conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.