Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 2, pp. 531538 (2003) 

Dark clouds on spheres and totally nonspherical bodies of constant breadthRené Brandenberg and David LarmanZentrum Mathematik, Technische Universität München, D80290 Munich, Germany, email: brandenb@ma.tum.de; Department of Mathematics, University College London, London WC1E 6BT, email: d.larman@ucl.ac.ukAbstract: In this paper, we show that for any dimension $d \ge 3$ there exists a body of constant breadth $C$, such that its projection onto any 2plane is nonspherical. We call such a body totally nonspherical. The circumradius of the projection of any totally nonspherical body $C$ of constant breadth onto any 2plane is bigger than the half diameter of $C$. Showing the existence of such a body extends results of Eggleston [E] and Weissbach [W], who showed it in the case $d=3$. [E] Eggleston, H. G.: Minimal universal covers in ${E^n$}. Israel J. Math. {\bf 16} (1963), 149155. [W] Weissbach, B.: {Über die senkrechten Projektionen regul{ä}rer Simplexe}. Beitr. Algebra Geom. {\bf 15} (1983), 3541. Keywords: radii, minimal projections, isoperimetric inequalities, dark clouds, constant breadth, constant width, nonspherical Full text of the article:
Electronic version published on: 1 Aug 2003. This page was last modified: 4 May 2006.
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