The covariogram and Fourier-Laplace transform in $\mathbb{C}^n$

The covariogram $g_{K}$ of a convex body $K$ in $\mathbb{R}^n$ is the function which associates to each $x\in\mathbb{R}^n$ 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 $\hat{1_K}$ in $\mathbb{R}^n$, a particular instance of the Phase Retrieval Problem. We connect the Covariogram Problem to two aspects of the Fourier transform $\hat{1_K}$ seen as a function in $\mathbb{C}^n$. The first connection is with the problem of determining $K$ from the knowledge of the zero set of $\hat{1_K}$ in $\mathbb{C}^n$. 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 $\hat{1_K}$. This connection also shows a link between the Covariogram Problem and the Pompeiu Problem in integral geometry.