The covariogram and Fourier-Laplace transform in \(\mathbb{C}^{n}\) (Q2827961)

The covariogram \(g_K\) of a convex body \(K\) is the function in \(\mathbb{R}^n\) defined by NEWLINE\[NEWLINEg_K(x)=vol_n\left(K \cap (K+x)\right),NEWLINE\]NEWLINE or, alternatively, as the convolution \(1_K * 1_{-K}\). The Covariogram Problem consists on determining the body \(K\) from its covariogram. The answer is positive on the plane, and for convex polytopes in \(\mathbb{R}^3\), but it is open for general convex bodies in \(\mathbb{R}^3\), and there are counterexamples for unique determination in dimension 4 and higher, as well as positive results in certain classes of polytopes. The convolution definition of \(g_K\) allows to reformulate the problem as a particular case of the \textit{phase retrieval problem}: Does \(|\widehat{1_K}(x)|\) determine the body \(K\) up to translations and reflections?NEWLINENEWLINEThe main result of this paper is a positive answer to the Covariogram Problem for convex bodies with certain regularity and positive Gaussian curvature, extending a result by \textit{T. Kobayashi} [J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 3, 389--478 (1989; Zbl 0701.32008); ``Convex domains and Fourier transform on spaces of constant curvature'', in: Lecture Notes of the UNESCO-CIMPA School on ``Invariant differential operators on Lie groups and homogeneous spaces''. Wuhan, China: WuHan University. 112 p. (1991), \url{http://www.ms.u-tokyo.ac.jp/~toshi/texdvi/nulv.dvi}], who had considered \(C^\infty_+\) bodies. The proof follows Kobayashi's ideas regarding the asymptotics of the zero set \(\mathcal{Z}(K)=\{\zeta \in \mathbb{C}^n: \widehat{1_K}(\zeta)=0\}\).NEWLINENEWLINETheorem 1. Let \(n \geq 2\), and define \(r(n)=8\) for \(n=2,4, 6\), \(r(n)=9\) for \(n=3, 5, 7\) and \(r(n)=\left[(n-1)/2\right]+5\) for \(n \geq 8\). Let \(H,K\) be convex bodies in \(\mathbb{R}^n\) of class \(C^{r(n)}_+\). If \(g_H=g_K\), then \(H\) and \(K\) coincide, up to translations and reflections.NEWLINENEWLINEThe same ideas allow the author to prove a similar result for the cross-covariogram of two planar convex bodies with high enough regularity.