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Extremal lattices of convex bodies

1953; Cambridge University Press; Volume: 49; Issue: 1 Linguagem: Inglês

10.1017/s0305004100028188

1469-8064

H. P. F. Swinnerton-Dyer,

Point processes and geometric inequalities

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 L i ( 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 L i . 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.