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

Abstract

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.