Confirmation of Matheron's conjecture on the covariogram of a planar convex body

2007/11/05 by Gennadiy Averkov, Averkov, Gennadiy, Gabriele Bianchi +1 · 1 citation
Materials Science · Mathematics · Medicine · Physics and Astronomy · #60D05 #Astronomical and nuclear sciences #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Morphological variations and asymmetry #Pharmacological Effects of Medicinal Plants #Point processes and geometric inequalities #Probability (math.PR) #Quasicrystal Structures and Properties #math.CA #math.MG #math.PR #msc:60D05

paper · pdf · doi:10.48550/arxiv.0711.0572

12 pages, 7 figures

openalex publication_date 2007/11/05 · arxiv created 2007/11/30 · arxiv updated 2011/11/09 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K 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.

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