A set of dual-extreme lattices
A lattice \(L\) in \(n\)-dimensional euclidean space is called dual-extreme if the geometric mean of the Hermite numbers of \(L\) and its dual \(L^*\) attains a local maximum. Following the classical method of Voronoi, \textit{A.-M. Bergé} and \textit{J. Martinet} [J. Number Theory 32, No. 1, 14-42 (1989; Zbl 0677.10022)] have characterized such lattices as being dual-perfect and dual-eutactic in the appropriate sense. As a consequence, if \(L\) is extreme and \(L^*\) is eutactic, then \(L\) is dual-extreme. However, for any even \(n\geq 8\), this paper gives an example where \(L\) is dual-extreme, neither \(L\) nor \(L^*\) is perfect, and only one of them is eutactic.
scientific article; zbMATH DE number 1132892