Application of an idea of Voronoĭ to lattice packing
The famous result of Voronoĭ states that a lattice packing of balls has locally maximum density if and only if it is eutactic and perfect. As a generalization, for any smooth convex ball \(c\) concepts of \(C\)-semi-eutaxy, \(C\)-eutaxy, and \(C\)-perfection are intriduced. These conceptions are connected with semi-stationarity, stationarity, and local ultra-maximality of lattice packing density for \(C\). For example, it is proved that the packing of \(C\) by the lattice \(L\) has local ultra-maximum at \(L\) if and only if \(L\) is eutactic and perfect with respect to \(C\). Similar results also proved for the product of packing densities of smooth convex body \(C\) and its polar body \(C^*\).
scientific article; zbMATH DE number 6342886
scientific article