The solution of the covariogram problem for plane \({\mathcal C}_+^2\) convex bodies
For a convex body \(K\) in \(\mathbb{R}^n\), the geometric covariogram \(g_K\) is the function assigning to \(x\in\mathbb{R}^n\) the \(n\)-dimensional Lebesgue measure of the intersection \(K\cap(K+ x)\). The covariogram problem, going back to G. Matheron, asks whether the geometric covariogram of \(K\) determines \(K\), among all convex bodies in \(\mathbb{R}^n\), up to translation and reflection. An affirmative answer in the case when \(n= 2\) and \(K\) is a polygon was given by \textit{W. Nagel} [J. Appl. Probab. 30, No. 3, 730--736 (1993; Zbl 0781.60018)]. Here, the authors succeed in giving a positive answer for planar convex bodies of class \(C^2\) with non-vanishing curvature. They also extend to \(n= 2\) the fact, previously known for \(n\geq 3\), that the covariogram problem has a positive answer for most convex bodies, in the Baire category sense.
scientific article; zbMATH DE number 2052862