To begin with, Minkowski's theorem establishes a relation between symmetric convex sets and integer points; we might as well say, between any lattice and any Banach space norm in n dimensions. The topic therefore belongs properly to a sort of affine geometry simplification of the theory of quadratic forms (Hilbert space norms in relation to lattices). To relax the convexity technique in a non-trivial way may be technically difficult.

The theoretical foundations can be considered as dealing with the **space of lattices** in n dimensions, which is *a priori* the coset space GL_{n}(R)/GL_{n}(Z). This isn't very easy to deal with directly (it is an example for the theory rather of arithmetic groups). One foundational result is *Mahler's compactness theorem* describing the relatively compact subsets (the coset space is non-compact, as can be seen already in the case n = 2, where there are *cusps*).

One can say that the geometry of numbers takes on some of the work that continued fractions do, for diophantine approximation questions in two or more dimensions - there is no straightforward generalisation.