JournalsjemsVol. 11 , No. 6DOI 10.4171/jems/179

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

Abstract

The covariogram gK of a convex body K in Ed is the function which associates to each xEd 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.