# 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 $g_{K}$ 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 $g_{K}$ 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.

## Cite this article

Gennadiy Averkov, Gabriele Bianchi, Confirmation of Matheron's conjecture on the covariogram of a planar convex body. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1187–1202

DOI 10.4171/JEMS/179