We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.

By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.

A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.