Confirmation of Matheron's conjecture on the covariogram of a planar convex body
By G. Averkov, Gabriele Bianchi. 2007
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The covariogramgK of a convex bodyK in E d is the function which associates to each x2 E d the volume of the intersection ofK withKCx. In 1986 G. Matheron conjectured that for dD 2 the covariogramgK determinesK within the class of all planar convex bodies, up to transla- tions and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as 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. In this paper we confirm Matheron's conjecture completely.
Subfields: Applied Mathematics, Geometry and Topology, Molecular Biology
View Paper: https://www.semanticscholar.org/paper/323680931529f8813ac0246fe57de9a6fc4e069e
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In Applied Mathematics
In Geometry and Topology
In Molecular Biology
Citations in the corpus
- COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS (2011, background): “As an important example, Averkov and Bianchi have recently established Matheron’s conjecture: up to a translation and a reflection, convex bodies of R2, that is compact convex sets with non-empty interior, are fully determined by their covariogram (see (Averkov and Bianchi, 2009) and the references ”