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

### Gennadiy Averkov

Otto-von-Guericke-Universität, Magdeburg, Germany### Gabriele Bianchi

Università degli Studi di Firenze, Italy

## Abstract

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