The solution of the covariogram problem for plane \({\mathcal C}_+^2\) convex bodies
type: Item
instance of: scholarly article
title: The solution of the covariogram problem for plane \({\mathcal C}_+^2\) convex bodies
zbMATH Open document ID: 1047.52002
author: Gabriele Bianchi
author: Fausto Segala
author: Aljoša Volčič
published in: Journal of Differential Geometry
publication date: 2004-03-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 2052862
zbMATH Keywords: covariogram
zbMATH Keywords: determination of convex bodies
reviewed by: Rolf Schneider
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090351101
OpenAlex ID: W2182574588
Wikidata QID: Q115201785
DOI: 10.4310/JDG/1090351101
description: scientific article; zbMATH DE number 2052862