Reprint: On lattice points in $n$-dimensional star bodies (1946)

Abstract

Let $F(X)=F(x_1,\dots ,x_n)$ be a continuous non-negative function of $X=(x_1,\dots ,x_n)$ that satisfies $F(tX)=|t|F(X)$ for all real numbers $t$. The set $K$ in $n$-dimensional Euclidean space ${\mathbb{R}}^n$ defined by $F(X)⩽1$ is called a star body. In this paper, Mahler studies the lattices $\Lambda$ in ${\mathbb{R}}^n$ which are of minimum determinant and have no point except $(0,\dots ,0)$ 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.

Reprint of the author's paper [Proc. R. Soc. Lond., Ser. A 187, 151--187 (1946; Zbl 0060.11710)].