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The determination of convex bodies from the mean of random sections

Published online by Cambridge University Press:  24 October 2008

Paul Goodey  and
Wolfgang Weil
Paul Goodey
Affiliation:
Mathematics Department, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.
Wolfgang Weil
Affiliation:
Mathematisches Institut II, Universität Karlsruhe, Englerstrasse 2, 7500 Karlsruhe 1, Germany
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Random sectioning of particles (compact sets in ℝ3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [23] or the book [20]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection Xk = K ∩ E is a (k-dimensional, k = 1 or 2) random set. It is clear that the distribution of Xk determines K uniquely and that if E1,…, En are such flats, the most natural estimator for K would be the convex hull

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 419 - 430
Copyright
Copyright © Cambridge Philosophical Society 1992

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