Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-08T09:11:17.188Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Adler, R. J. and Pyke, R. (1991). Problem 91–3. Inst. Math. Statist. Bull. 20, 409.Google Scholar
Adler, R. J. and Pyke, R. (1997). Scanning Brownian processes. Adv. Appl. Prob. 29, 295–326.Google Scholar
Averkov, G. and Bianchi, G. (2007). Retrieving convex bodies from restricted covariogram functions. Preprint. Available at http://www.arxiv.org/abs/math/0702892.Google Scholar
Baake, M. and Grimm, U. (2007). Homometric model sets and window covariograms. Z. Kristallographie 222, 54–58.Google Scholar
Bianchi, G. (2002). Determining convex polygons from their covariograms. Adv. Appl. Prob. 34, 261–266.Google Scholar
Bianchi, G. (2005). Matheron's conjecture for the covariogram problem. J. London Math. Soc. (2) 71, 203–220.Google Scholar
Bianchi, G. (2005). Some open problems regarding the determination of a set from its covariogram. Le Matematiche (Catania) 40, 247–257.Google Scholar
Bianchi, G. (2006). The covariogram determines three-dimensional convex polytopes. Preprint.Google Scholar
Bianchi, G., Segala, F. and Volčič, A. (2002). The solution of the covariogram problem for plane C2 convex bodies. J. Differential Geometry 60, 177–198.Google Scholar
Cabo, A. and Baddeley, A. (2003). Estimation of mean particle volume using the set covariance function. Adv. Appl. Prob. 35, 27–46.CrossRefGoogle Scholar
Enns, E. G. and Ehlers, P. F. (1978). Random paths through a convex region. J. Appl. Prob. 15, 144–152.Google Scholar
Enns, E. G. and Ehlers, P. F. (1988). Chords through a convex body generated from within an embedded body. J. Appl. Prob. 25, 700–707.CrossRefGoogle Scholar
Enns, E. G. and Ehlers, P. F. (1993). Notes on random chords in convex bodies. J. Appl. Prob. 30, 889–897.Google Scholar
Gage, M. and Hamilton, R. S. (1986). The heat equation shrinking convex plane curves. J. Differential Geometry 23, 69–96.Google Scholar
Gardner, R. J. (1995). Geometric Tomography. (Encyclopaedia Math. Appl. 58). Cambridge University Press.Google Scholar
Goodey, P., Schneider, R. and Weil, W. (1997). On the determination of convex bodies by projection functions. Bull. London Math. Soc. 29, 82–88.Google Scholar
Gruber, P. M. (1993). Baire categories in convexity. In Handbook of Convex Geometry, Vol. A, B, North-Holland, Amsterdam, pp. 1327–1346.Google Scholar
Hug, D. and Schneider, R. (2002). Stability results involving surface area measures of convex bodies. Rend. Circ. Mat. Palermo (2) Suppl. 21–51.Google Scholar
Mallows, C. L. and Clark, J. M. C. (1970). Linear-intercept distributions do not characterize plane sets. J. Appl. Prob. 7, 240–244.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Matheron, G. (1986). Le covariogramme géometrique des compacts convexes des R2. Tech. Rep. 2/86, Centre de Géostatistique, Ecole des Mines de Paris.Google Scholar
Nagel, W. (1991). Das geometrische Kovariogramm and verwandte Grössen zweiter Ordnung. Habilitationsschrift, Friedrich-Schiller-Universität Jena.Google Scholar
Nagel, W. (1993). Orientation-dependent chord length distributions characterize convex polygons. J. Appl. Prob. 30, 730–736.Google Scholar
Santaló, L. A. (2004). Integral Geometry and Geometric Probability, 2nd edn. Cambridge University Press.Google Scholar
Schmitt, M. (1993). On two inverse problems in mathematical morphology. In Mathematical Morphology in Image Processing (Opt. Eng. 34), Dekker, New York, pp. 151–169.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopaedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schneider, R. (1994). Polytopes and Brunn–Minkowski theory. In Polytopes: Abstract, Convex and Computational, Kluwer, Dordrecht, pp. 273–299.Google Scholar
Schneider, R. (1998). On the determination of convex bodies by projection and girth functions. Results Math. 33, 155–160.Google Scholar
Serra, J. (1984). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Skiena, S. S. (2004). Geometric reconstruction problems. In Handbook of Discrete and Computational Geometry, 2nd edn, Boca Raton, FL, pp. 665–676.Google Scholar