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Retrieving convex bodies from restricted covariogram functions

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  01 July 2016

Gennadiy Averkov  and
Gabriele Bianchi
Gennadiy Averkov*
Affiliation:
University of Magdeburg
Gabriele Bianchi*
Affiliation:
Università di Firenze
*
∗ Postal address: Faculty of Mathematics, University of Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany. Email address: gennadiy.averkov@googlemail.com
∗∗ Postal address: Department of Mathematics, Università di Firenze, Viale Morgagni 67a, 50134 Firenze, Italy. Email address: gabriele.bianchi@unifi.it
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Abstract

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The covariogram gK(x) of a convex body K ⊆ Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x, where Ed denotes the Euclidean d-dimensional space. Matheron (1986) asked whether gK determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d ≥ 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determination. We indicate some subsets of the support of the covariogram, with arbitrarily small Lebesgue measure, such that the covariogram, restricted to those subsets, identifies certain geometric properties of the body. These results are more precise in the planar case, but some of them, both positive and negative ones, are proved for bodies of any dimension. Moreover some results regard most convex bodies, in the Baire category sense. Another purpose is to extend the class of convex bodies for which Matheron's conjecture is confirmed by including all planar convex bodies possessing two nondegenerate boundary arcs being reflections of each other.

Keywords

Convex bodyconvex polytopecovariogramgenericity resultgeometric tomographyset covariancequasicrystal

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Secondary: 52A22: Random convex sets and integral geometry 52A38: Length, area, volume 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 613 - 629
Copyright
Copyright © Applied Probability Trust 2007 

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