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

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.