The covariogram determines three-dimensional convex polytopes

The cross covariogram gK,L of two convex sets K,L⊂Rn is the function which associates to each x∈Rn the volume of the intersection of K with L+x. The problem of determining the sets from this function is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. The two main results of this paper are that gK,K determines three-dimensional convex polytopes K and that gK,L determines both K and L when K and L are convex polyhedral cones satisfying certain assumptions. These results settle a conjecture of G. Matheron in the class of convex polytopes. Further results regard the known counterexamples in dimension n⩾4.