Lattice octahedra and sums of powers of linear forms

In “-dimensional real space, let u₁…, u_nbe n linearly independent points of a lattice L. Assume that the closed octahedron formed by the convex hull of the points ±u₁…, ±u_ncontains no points of L other than the origin and the octahedron vertices. An old problem, solved by Minkowski for n = 2 and 3, concerns the possible values of the index I in L of the sublattice generated by the u₁We prove that there exist lattices L such that I ≥ n!/2ⁿfor n ≥ 5. This result is used to prove the existence of n linear forms in n variables with determinant D, the sum of whose absolute values is, asymptotically, at least n|D|^1/n/2e. Similar results are proved for the sum of the pth powers of the absolute values of n linear forms, where p is any fixed real number not less than 1. For quadratic forms (p = 2) our result matches a classical theorem of Minkowski; for p ≠ 2 our result is new.