Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-18T08:23:27.930Z Has data issue: false hasContentIssue false
  • English
  • Français

COVARIOGRAM OF NON-CONVEX SETS

Part of: General convexity Geometric probability and stochastic geometry Harmonic analysis in several variables

Published online by Cambridge University Press:  21 June 2010

Carlo Benassi ,
Gabriele Bianchi  and
Giuliana D’Ercole
Carlo Benassi
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, via Campi 213/B, Modena, I-41100, Italy (email: benassi.carlo@unimore.it)
Gabriele Bianchi
Affiliation:
Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/A, Firenze, I-50134, Italy (email: gabriele.bianchi@unifi.it)
Giuliana D’Ercole
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, via Campi 213/B, Modena, I-41100, Italy (email: giulianadercole@libero.it)
Get access

Abstract

The covariogram of a compact set A⊂ℝn is the function that to each x∈ℝn associates the volume of A∩(A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than nine points that have equal discrete covariograms.

MSC classification

Secondary: 60D05: Geometric probability and stochastic geometry 52A10: Convex sets in $2$ dimensions (including convex curves) 52A22: Random convex sets and integral geometry 52A38: Length, area, volume 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 267 - 284
Copyright
Copyright © University College London 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adler, R. J. and Pyke, R., Problem 91-3. The Institute of Mathematical Statistics Bulletin 20 (1991), 409.Google Scholar
[2]Averkov, G. and Bianchi, G., Confirmation of Matheron’s conjecture on the covariogram of planar convex bodies. J. Eur. Math. Soc. (JEMS) 11 (2009), 1187–1202.CrossRefGoogle Scholar
[3]Benassi, C. and D’Ercole, G., An algorithm for reconstructing a convex polygon from its covariogram. Rend. Istit. Mat. Univ. Trieste 39 (2007), 457–476.Google Scholar
[4]Bianchi, G., The covariogram determines three-dimensional convex polytopes. Adv. Math. 220 (2009), 1771–1808.CrossRefGoogle Scholar
[5]Cohn, D. L., Measure Theory, Birkhäuser (Boston, MA, 1980).CrossRefGoogle Scholar
[6]Daurat, A., Gérard, Y. and Nivat, M., The chords’ problem. FUN with algorithms (Elba, 1998). Theoret. Comput. Sci. 282 (2002), 319–336.CrossRefGoogle Scholar
[7]Daurat, A., Gérard, Y. and Nivat, M., Some necessary clarifications about the chords’ problem and the partial digest problem. Theoret. Comput. Sci. 347 (2005), 432–436.CrossRefGoogle Scholar
[8]Gardner, R. J., Gronchi, P. and Zong, C., Sums, projections, and selections of lattice sets, and the discrete covariogram. Discrete Comput. Geom. 34 (2005), 391–409.CrossRefGoogle Scholar
[9]Lešanovský, A., Rataj, J. and Hojek, S., 0-1 sequences having the same number of (1-1) couples of given distances. Math. Bohem. 117 (1992), 271–282.CrossRefGoogle Scholar
[10]Matheron, G., Random Sets and Integral Geometry, Wiley (New York, NY, 1975).Google Scholar
[11]Matheron, G., Le covariogramme géometrique des compacts convexes de ℝ2. Technical ReportN-2/86/G, Centre de Géostatistique, Ecole Nationale Supérieure des Mines de Paris, 1986.Google Scholar
[12]Rosenblatt, J. and Seymour, P. D., The structure of homometric sets. SIAM J. Algebraic Discrete Methods 3 (1982), 343–350.CrossRefGoogle Scholar
[13]Schneider, R., Convex Bodies: the Brunn–Minkowski Theory, Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar