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

TL;DR

This paper proves Matheron's conjecture that the covariogram uniquely determines a planar convex body up to translation and reflection, resolving a longstanding problem with implications in stochastic geometry, Fourier analysis, and material science.

Contribution

The paper provides a complete proof confirming that the covariogram uniquely identifies planar convex bodies, advancing understanding in geometric tomography and related fields.

Findings

Confirmed Matheron's conjecture for d=2

Established the uniqueness of covariogram in determining convex bodies

Implications for phase retrieval and quasicrystal analysis

Abstract

The covariogram g_K of a convex body K in E^d is the function which associates to each x in 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.