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 GLn(R)/GLn(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.