An estimate for the number of minimal vectors of point lattices. (Q1425505)

Let \(s(\Gamma^n)\) be the number of pairss \(\pm \bar m\) of minimal vectors of a Euclidean lattice \(\Gamma^n\) of rank \(n\). Also let \(s_n = \max s(\Gamma^n)\), where the maximum is taken over all lattices \(\Gamma^n\) of rank \(n\). For every \(n\geq 3\) an estimate of the value \(s_n\) \((s_n\geq 5\cdot 2^{n-3}+2)\) is obtained which yet for \(n\geq 4\) is somewhat better than the known Voronoi estimate \((s_n\geq 2^n-1)\) [\textit{G. F. Voronoi}, J. Reine Angew. Math. 134, 198--287 (1908; JFM 39.0274.01)].