Extremal lattices of convex bodies

Abstract

Let C be a bounded closed convex body in n dimensions, symmetric about the origin. Any lattice Λ containing the origin but no other interior point of C is called admissible. There is a positive lower bound Δ(C) for the determinants of admissible lattices (since the origin is inside C); and any admissible lattice with determinant Δ(C) is called critical. Suppose that Λ is any admissible lattice, with determinant d(Λ). We may define A by a fixed set of generating points Li (i = 1,2, …, n); and we shall say that a lattice Λ′ lies in a small neighbourhood of Λ if Λ′ can be generated by a set of points L′i (i = 1,2, …, n) each of which lies in a small neighbourhood of the corresponding Li. We shall call Λ extremal if in a sufficiently small neighbourhood of Λ there are no admissible lattices Λ′ with d(Λ′) < d(Λ). Thus all critical lattices are extremal.