Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 44, No. 2, pp. 531-538 (2003)

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Dark clouds on spheres and totally non-spherical bodies of constant breadth

René Brandenberg and David Larman

Zentrum Mathematik, Technische Universität München, D--80290 Munich, Germany, e-mail:; Department of Mathematics, University College London, London WC1E 6BT, e-mail:

Abstract: 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 2-plane is non-spherical. We call such a body totally non-spherical. The circumradius of the projection of any totally non-spherical body $C$ of constant breadth onto any 2-plane 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), 149--155.

[W] Weissbach, B.: ber die senkrechten Projektionen regul{ä}rer Simplexe}. Beitr. Algebra Geom. {\bf 15} (1983), 35--41.

Keywords: radii, minimal projections, isoperimetric inequalities, dark clouds, constant breadth, constant width, non-spherical

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Electronic version published on: 1 Aug 2003. This page was last modified: 4 May 2006.

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