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 of a convex body in is the function which associates to each the volume of the intersection of with . In 1986 G. Matheron conjectured that for the covariogram determines 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