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

Source abstract

The covariogram g_K of a convex body K in \mathbb E^d is the function which associates to each x\in \mathbb 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. 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.