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Orientation-dependent chord length distributions characterize convex polygons

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  14 July 2016

W. Nagel
W. Nagel*
Affiliation:
University of Jena
*
∗ Postal address: Mathematische Fakultät, Friedrich-Schiller-Universität Jena, D-07740 Jena, Germany.
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Abstract

It is shown that the convex polygons are uniquely determined (up to translation and reflection) by their covariograms. The covariogram can be represented by the ‘orientation-dependent chord length distribution', i.e. the distribution of the length of chords which are generated by random lines parallel to fixed directions. Thus the result contributes to answer Blaschke's question about the content of information comprised in chord length distributions.

Keywords

COVARIOGRAMGEOMETRIC PROBABILITYINTEGRAL GEOMETRYCONVEX POLYGONS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A10: Convex sets in $2$ dimensions (including convex curves) 52A22: Random convex sets and integral geometry

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 730 - 736
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Adler, R. J. and Pyke, R. (1991) Problem 91-93. IMS Bull. 20, 407.Google Scholar
[2] Bonnesen, T. and Fenchel, W. (1934) Theorie der konvexen Körper. Ergebnisse der Mathematik und ihrer Grenzgebiete 3(1), 1–172.Google Scholar
[3] Enns, E. G. and Ehlers, P. F. (1978) Random paths through a convex region. J. Appl. Prob. 15, 144–152.Google Scholar
[4] Gates, J. (1982) Recognition of triangles and quadrilaterals by chord length distribution. J. Appl. Prob. 19, 873–879.Google Scholar
[5] Leichtweiss, K. (1980) Konvexe Mengen. VEB Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
[6] Mallows, C. L. and Clark, J. M. (1970) Linear intercept distributions do not characterize plane sets. J. Appl. Prob. 7, 240–244.Google Scholar
[7] Mathéron, G. (1965) Les variables régionalisées et leur estimation. Masson, Paris.Google Scholar
[8] Mathéron, G. (1975) Random Sets and Integral Geometry, Wiley, New York.Google Scholar
[9] Nagel, W. (1989) Geometric covariograms for pairs of parallel j-planes. In: Geobild '89, ed. Hübler, A. et al. Akademie-Verlag, Berlin.Google Scholar
[10] Nagel, W. (1991) Das geometrische Kovariogramm und verwandte Größen zweiter Ordnung. Habilitationsschrift, Friedrich-Schiller-Universität Jena.Google Scholar
[11] Rogers, C. A. and Shephard, G. C. (1957) The difference body of a convex body. Arch. Math. 8, 220–233.Google Scholar
[12] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
[13] Schneider, R. (1970) Eine Verallgemeinerung des Differenzenkörpers. Monatsh. Math. 74, 258–272.Google Scholar
[14] Waksman, P. (1985) Plane polygons and a conjecture of Blaschke's. Adv. Appl. Prob. 17, 774–793.Google Scholar