Let be a continuous non-negative function of that satisfies for all real numbers . The set in -dimensional Euclidean space defined by is called a star body. In this paper, Mahler studies the lattices in which are of minimum determinant and have no point except inside . He investigates how many points of such lattices lie on, or near to, the boundary of , and considers in detail the case when admits an infinite group of linear transformations into itself.
Reprint of the author's paper [Proc. R. Soc. Lond., Ser. A 187, 151--187 (1946; Zbl 0060.11710)].