Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The determination of convex bodies from the mean of random sections

Paul Goodeya1 and Wolfgang Weila2

a1 Mathematics Department, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.

a2 Mathematisches Institut II, Universität Karlsruhe, Englerstrasse 2, 7500 Karlsruhe 1, Germany

Random sectioning of particles (compact sets in xs211D3 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 xs2229 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

S0305004100071085_eqnU1

(Received February 13 1991)

(Revised January 07 1992)