Small, Christopher G. A counterexample to a conjecture on random shadows. (English) Zbl 0772.52007 Can. J. Stat. 20, No. 4, 463-468 (1992). Let \(\pi(K)\) be the orthographic shadow of a body \(K\) on the \(x_ 1\), \(x_ 2\) plane. Two compact convex sets \(K_ 1\) and \(K_ 2\) will be said to have the same shadow distribution if there exist two random vectors \(a_ 1\) and \(a_ 2\) lying in the \(x_ 1\), \(x_ 2\) plane such that the random sets \(\pi(u(K_ 1)) + a_ 1\) and \(\pi(u(K_ 2)) + a_ 2\) have the same distribution. Then P. M. Cullagh, in a private communication to the author, has conjectured that any two compact convex \(K_ 1\) and \(K_ 2\) which have the same shadow distribution are congruent. The author constructs a counterexample of this conjecture, and some applications to stereology are discussed. Reviewer: L.A.Santaló (Buenos Aires) MSC: 52A22 Random convex sets and integral geometry (aspects of convex geometry) 60D05 Geometric probability and stochastic geometry Keywords:random shadows; integral geometry; orthographic projection; geometric probability; convex sets; stereology PDFBibTeX XMLCite \textit{C. G. Small}, Can. J. Stat. 20, No. 4, 463--468 (1992; Zbl 0772.52007) Full Text: DOI References: [1] Betke, Estimating the sizes of convex bodies from projections, J. London Math. Soc, 27 pp 525– (1983) · Zbl 0487.52005 [2] Bol, über Eikcirper mil Vieleckschatten, Math. Z. 48 pp 227– (1942) [3] Bonneson, Om Minkowski’s uligheder fur konvexer legemer, Mat. Tidsskr. B, 80 (1926) [4] Chakerian, Is a body spherical if all its projections have the same I.Q.?, Amer. Math. Monthly 77 pp 989– (1970) [5] Firey, Blaschke sum of convex bodies and mixed bodies pp 94– (1967) · Zbl 0153.51902 [6] Godambe, A unified theory of sampling from finite populations, J. Roy. Statist. Soc. Ser. B, 17 pp 269– (1955) · Zbl 0067.11406 [7] Keville, Characterizations of dimensions of ellipsoidal microparticles via electron microscopy, J. Microscopy 142 pp 327– (1986) [8] Klee, Some characterizations of convex polyhedra, Ada Math. 102 pp 79– (1959) · Zbl 0094.16802 [9] Klee, On a conjecture of Lindenstaruss, Israel J. Math. 1 pp 1– (1963) [10] Kuz’minykh, Recovery of a convex body from the set of its projections, Sibirsk. Mat. Zh. 25 pp 145– (1984) [11] Matheron, Random Sets and Integral Geometry (1975) [12] Shephard, Twenty problems on convex polyhedra II, Math. Gaz. 52 pp 359– (1968) · Zbl 0161.41604 [13] Small, Reconstructing convex bodies from random projected images. Canad. J. Statist., to appear. Underwood, E.E. (1971). The stereology of projected images., J. Microscopy 95 pp 25– (1991) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.