Minimal pairs of convex bodies in two dimensions

In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hormander-Radstrom lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝ n is contained in another convex body. This generalizes a theorem of Weil ( cf . [10]).