The cross covariogram of a pair of polygons determines both polygons, with a few exceptions

The cross covariogram gK,L of two convex sets K and L in Rn is the function which associates to each x∈Rn the volume of K∩(L+x). We prove that when K and L are either convex polygons or planar convex cones, gK,L determines both K and L, up to a described family of exceptions. These results imply that, when K and L are in these classes, the information provided by the cross covariogram is so rich as to determine not only one unknown body, as required by Matheron's conjecture, but two bodies, with a few classified exceptions.