Geodesic
Confirmation of Matheron's conjecture on the covariogram of a planar convex body
Journal of the European Mathematical Society, Tome 11 (2009) no. 6, pp. 1187-1202.

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The covariogram gK​ of a convex body K in Ed is the function which associates to each x∈Ed the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram gK​ determines K within the class of all planar convex bodies, up to translations 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. 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.
DOI : 10.4171/jems/179
Classification : 60-XX, 42-XX, 52-XX, 00-XX
Keywords: Autocorrelation, covariogram, cut-and-project scheme, geometric tomography, image analysis, phase retrieval, quasicrystal, set covariance 
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     title = {Confirmation of {Matheron's} conjecture on the covariogram of a planar convex body},
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Gennadiy Averkov; Gabriele Bianchi. Confirmation of Matheron's conjecture on the covariogram of a planar convex body. Journal of the European Mathematical Society, Tome 11 (2009) no. 6, pp. 1187-1202. doi : 10.4171/jems/179. http://geodesic.mathdoc.fr/articles/10.4171/jems/179/

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