A lecture on the geometry of numbers of convex bodies

Abstract

This determinant is T^O if and only if the n points are linearly independent over R. Points with integral coordinates are called lattice points, and we use A to denote the lattice of all such lattice points. A is an Abelian group with n independent generators under addition. Every bounded set contains at most finitely many lattice points. We shall be concerned with the relation between A and convex bodies. Here a convex body K is to mean a bounded closed convex set in R which contains the origin as an interior point and is symmetric in 0. Important examples are the cube \x ^ 1 , • • • , \xn g l , the octahedron \xi + • • • +\xn g l , and the sphere # ? + • • • +xl^l. The volume of a convex body K is defined by