On lattice points in n -dimensional star bodies I. Existence theorems

Abstract

Let F ( X ) = F ( x 1 ,..., x n ) be a continuous non-negative function of X satisfying F ( tX ) = | t | F ( X ) for all real numbers t . The set K in n -dimensional Euclidean space R n defined by F ( X )⩽ 1 is called a star body. The author studies the lattices Λ in R n which are of minimum determinant and have no point except (0, ..., 0) inside K . He investigates how many points of such lattices lie on, or near to, the boundary of K , and considers in detail the case when K admits an infinite group of linear transformations into itself.