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Application of an idea of Voronoĭ to lattice packing. (English) Zbl 1297.05047
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 \(C^*\).

MSC:
05B40 Combinatorial aspects of packing and covering
11H06 Lattices and convex bodies (number-theoretic aspects)
11H31 Lattice packing and covering (number-theoretic aspects)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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