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

Abstract Let F(X) = F(x1,..., xn) 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 Rn defined by F(X)⩽ 1 is called a star body. The author studies the lattices Λ in Rn 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.