Minimal Determinants and Lattice Inequalities

Some results of P. McMullen on determinants of sublattices ofZ^d induced by rational subspaces are generalized to arbitrarylattices. As an application, we obtain an equality for the minimaldeterminants introduced by J. M. Wills, namely D_t(L) = D_d(L)D_d–1((L*).Using an inequality of Lagarias, Lenstra and Schnorr, we generalizetwo isoperimetric inequalities withlattice constraints by Bokowski,Hadwiger and Wills, and Hadwiger, respectively, to arbitrarylattices.