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

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

2007-11-05

arxiv.org/abs/0711.0572v2

Mathematics

Metric Geometry

12 pages, 7 figures

Scientific paper

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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