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

  • Kurt Mahler


Let F(X)=F(x1,,xn)F(X)=F(x_1,\ldots,x_n) be a continuous non-negative function of X=(x1,,xn)X=(x_1,\ldots,x_n) that satisfies F(tX)=tF(X)F(tX)=|t|F(X) for all real numbers tt. The set KK in nn-dimensional Euclidean space Rn\mathbb{R}^n defined by F(X)1F(X)\leqslant 1 is called a star body. In this paper, Mahler studies the lattices Λ\Lambda in Rn\mathbb{R}^n which are of minimum determinant and have no point except (0,,0)(0,\ldots,0) inside KK. He investigates how many points of such lattices lie on, or near to, the boundary of KK, and considers in detail the case when KK 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)].