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On the volume of parallel bodies: a probabilistic derivation of the Steiner formula

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Richard A. Vitale
Richard A. Vitale*
Affiliation:
University of Connecticut
*
* Postal address: Department of Statistics, Box U-120, University of Connecticut, Storrs, CT 06269–3120, USA.
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Abstract

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.

Keywords

RANDOM VOLUMESRANDOM CONVEX BODIES

MSC classification

Primary: 52A22: Random convex sets and integral geometry
Secondary: 52A38: Length, area, volume 52A39: Mixed volumes and related topics 60D05: Geometric probability and stochastic geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 97 - 101
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Supported in part by ONR Grant N00014-90-J-1641 and NSF Grant DMS-9002665.

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis, held at the Universidad Internacional Menendez Pelayo, Valencia, Spain on 21–24 September 1994.

References

[1] Artstein, Z. and Vitale, R. A. (1975) A strong law of large numbers for random compact sets. Ann. Prob. 3, 879–882.Google Scholar
[2] Eggleston, H. G. (1969) Convexity. Cambridge University Press.Google Scholar
[3] Hoeffding, W. (1961) The law of large numbers for U-statistics. University of North Carolina, Institute of Statistics Mimeo Series, No. 302.Google Scholar
[4] Ruben, H. (1979) The volume of an isotropic random parallelotope. J. Appl. Prob. 16, 84–94.Google Scholar
[5] Schneider, R. (1993) Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press.Google Scholar
[6] Serfling, R. A. (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York.Google Scholar
[7] Shepard, G. C. (1974) Combinatorial properties of associated zonotopes. Canad. J. Math. 26, 302–321.Google Scholar
[8] Vitale, R. A. (1991) Expected absolute random determinants and zonoids. Ann. Appl. Prob. 1, 293–300.Google Scholar
[9] Vitale, R. A. (1994) Convex-geometric methods for testing directional data. In preparation.Google Scholar