On Mordell's inverse problem in dimension three (Q1915697)

Let \(L\subset E^3\) be a lattice with \(\text{det } L>0\), and let \(\kappa (L)= \sup\{\text{vol} (P)/ 8 \text{det } L\}\), where the supremum is taken over all \(O\)-symmetric parallelpipeds \(P\) with faces parallel to the coordinate planes such that \(\{0\}\) is the only \(L\)-point in \(P\). P. Gruber conjectured in 1970 that the absolute minimum of \(\kappa\) is attained if \(L\) is the critical lattice \(L^*\) of the star body \(|x_1 x_2 x_3 |\leq 1\). The author proves this and shows that this minimum is isolated.