introduced by J. M. Wills, namely Dt(L) = Dd(L)Dd_((L*). Using an inequality of ..... result of Conway and Thompson yields
MINIMAL DETERMINANTS AND LATTICE INEQUALITIES UWE SCHNELL
ABSTRACT Some results of P. McMullen on determinants of sublattices of Z d induced by rational subspaces are generalized to arbitrary lattices. As an application, we obtain an equality for the minimal determinants introduced by J. M. Wills, namely Dt(L) = Dd(L)Dd_((L*). Using an inequality of Lagarias, Lenstra and Schnorr, we generalize two isoperimetric inequalities with lattice constraints by Bokowski, Hadwiger and Wills, and Hadwiger, respectively, to arbitrary lattices.
1. Introduction d
In the following, let E , d^2, denote the Euclidean d-space and let Jz?d denote the set of lattices LczEd with detL # 0. Let Le£fd be a lattice, and let E be an rdimensional linear subspace of Ed with a linear basis containing only vectors of L. Further, let E be the orthogonal complement subspace. It is known that E has a basis containing only vectors of L* (for example, [4, Lemma 1.1]), where L* denotes the dual lattice of L. Let A = .En Land A = E(]L*. From [4, Lemma 1.1], II = L*/Eand n = L/E are lattices in E and E respectively, where / denotes the orthogonal projection. P. McMullen [6] proved, for the special case of the integer lattice L = Z d , det(A) = det(A), det(II) = det(n).
(1) (2)
For the applications given in Sections 3 and 4, we need some further definitions. Let Jfd denote the set of compact convex sets Kc: Ed. For a KeJfd, let V and F denote its volume and surface area (for simplicity we use V and F also for d = 2, where they denote area and perimeter). For KeJfd and LeJ?d, let G{K,L) = card(ATlL) denote the lattice point enumerator. For Le££d, let CL denote its Dirichlet-Voronoi-cell (DV-cell, compare, for example, [2]). For our purposes, we delete from any pair of opposite facets of CL exactly one, so that L + CL = Ed up to a (d— 2)-set (without change of notation). For the special case of the integer lattice L = Zd, it was shown in [1] that, if KeXd, then G{K)>V{K)-\F{K),
(3)
and this inequality is tight. Inequality (3) can be interpreted as an isoperimetric inequality with lattice constraints, and in fact the proof of (3) is essentially based on Hadwiger's isoperimetric inequality for lattice-periodic sets [3]. For this we need more definitions (compare [3]). A point set A is called L-periodic if A +g = A for each geL. We further assume that A and its boundary dA are in the following sense locally measurable. A(]CL and Received 17 July 1991; revised 2 December 1991. 1991 Mathematics Subject Classification 52C07, 11H06. Bull. London Math. Soc. 24 (1992) 606-612
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dA n CL can be obtained as limits of polyhedral approximations of A. The set of L-periodic sets A c Ed with these properties is denoted by s# dL and the subset of polyhedra by 0>dL. In general, AesrfdL is unbounded and V(A) and F(A) are not defined. So for Aes#dL we introduce with Hadwiger [3] the volume density V(A) = V(A D CL) and the surface area density F(A) = F(dA 0 CL) (which are independent of the above-mentioned choice of facets). Although V(K), F(K) for a Ke Jfd and V(A), F(A) fora.nAes/1 are not quite the same, we use the same notation to underline the analogy between (3) and (4). Now Hadwiger showed that, for L = Za and arbitrary Aestfd4V(A)(\-V(A)), (4) and this inequality is tight. For both inequalities. (3) and (4), there is the question of generalization to arbitrary lattices Leif*. In [8], Wills introduced the minimal determinants Dt of a lattice L Dt{L) = min {det Lt \ Lt is an /-dimensional sublattice of L), i = 1,..., d, (5) and for i = 0 we set £>0 = 1. Obviously, Dt(L) exists and Dt(L) > 0. The conjectured generalizations of (3) and (4) are for Le
