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Absolute curvature measures for unions of sets with positive reach

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Jan Rataj
Jan Rataj
Affiliation:
Mathematical Institute, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic.
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Abstract

Absolute curvature measures for locally finite unions of sets with positive reach are introduced, extending the definition of Zähle [13] by taking into account the absolute value of the index function. It is shown that this definition differs from that of Matheron [5] and Schneider [12]. An intersection formula of Crofton type for absolute curvature measures is proved. The role of absolute curvature measures in geometric statistics is illustrated by an example.

MSC classification

Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 33 - 44
Copyright
Copyright © University College London 2002

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References

1.Baddeley, A. J.. Absolute curvature measures in integral geometry. Math. Proc. Cambridge Philos. Soc., 88 (1980), 45–58.Google Scholar
2.Federer, H.Curvature measures. Trans. Amer. Math. Soc., 93 (1959), 418–491.CrossRefGoogle Scholar
3.Federer, H.Geometric Measure Theory (Springer Verlag, Berlin, 1969).Google Scholar
4.Kiêu, K., Souchet, S. and Istas, J.. Precision of systematic sampling and transitive methods. J. Statist. Plan. Infer., 11 (1999), 263–279.CrossRefGoogle Scholar
5.Matheron, G.. Random Sets and Integral Geometry (J. Wiley, New York, 1975).Google Scholar
6.Rataj, J.. A translative integral formula for absolute curvature measures. Geom. Dedicata. 84 (2001), 245–252.Google Scholar
7.Rataj, J. and Zähle, M.. Curvatures and currents for unions of sets with positive reach, II. Ann. Glob. Anal. Geom. 20 (2001), 1–21.Google Scholar
8.Rataj, J. and Zähle, M.. A remark on mixed curvature measures for sets with positive reach. Beiträge Alg. Geom. 43 (2002), 171–179.Google Scholar
9.Rother, W. and Zähle, M.. A short proof of a principal kinematic formula for curvature measures. Trans. Amer. Math. Soc., 321 (1990), 547–558.Google Scholar
10.Rother, W. and Zähle, M.. Absolute curvature measures, II. Geom. Dedicata, 41 (1992), 229–240.CrossRefGoogle Scholar
11.Santaló, L. A.. Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications 1 (Addison-Wesley, Reading 1976).Google Scholar
12.Schneiden, R.. Parallelmengen mit Vielfachheit und Steiner-Formeln. Geom. Dedicata, 9 (1980). 111–127.Google Scholar
13.Zähle, M.. Absolute curvature measures. Math. Nachr., 140 (1989), 83–90.Google Scholar
14.Zähle, M.. Nonosculating sets of positive reach. Geom. Dedicata, 76 (1999), 183–187.CrossRefGoogle Scholar