The covariogram and Fourier-Laplace transform in C^n

The covariogram g K of a convex body K in R n is the function which associates to each x ∈ R n the volume of the intersection of K with K + x. Determining K from the knowledge of g K is known as the Covariogram Problem. It is equivalent to determining the characteristic function 1 K of K from the modulus of its Fourier transform 1 K in R n , a particular instance of the Phase Retrieval Problem. We connect the Covariogram Problem to two aspects of the Fourier transform 1 K seen as a function in C^n . The first connection is with the problem of determining K from the knowledge of the zero set of 1 K in C n . To attack this problem T. Kobayashi studied the asymptotic behavior at infinity of this zero set. We obtain this asymptotic behavior assuming less regularity on K and we use this result as an essential ingredient for proving that when K is sufficiently smooth and in any dimension n, K is determined by g K in the class of sufficiently smooth bodies. The second connection is with the irreducibility of the entire function 1 K . This connection also shows a link between the Covariogram Problem and the Pompeiu Problem in integral geometry.