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The covariogram determines three-dimensional convex polytopes

By Gabriele Bianchi


The cross covariogram gK,L of two convex sets K, L ⊂ R n is the function which associates to each x ∈ R n the volume of the intersection of K with L + x. The problem of determining the sets from their covariogram 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. We also introduce and study the notion of synisothetic polytopes. This concept is related to the rearrangement of the faces of a convex polytope

Year: 2008
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