The covariogram problem

The covariogram gX of a measurable set X in Rn is the function which associates to each x ∈ Rn the measure of the intersection of X with X + x. We are interested in understanding what information about a set can be obtained from its covariogram. Matheron asked whether a convex body K is determined from the knowledge of gK, and this is known as the covariogram problem. The covariogram appears in very different contexts, and the covariogram problem can be rephrased in different terms. For instance, it is equivalent to determining the characteristic function 1K of K from the modulus of its Fourier transform 1K in Rn, a particular instance of the phase retrieval problem. The covariogram problem has also a discrete counterpart. We survey the known results and the methods.