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On the three-dimensional Blaschke-Lebesgue problem

Authors: Henri Anciaux and Brendan Guilfoyle
Journal: Proc. Amer. Math. Soc. 139 (2011), 1831-1839
MSC (2010): Primary 52A40, 52A15
Published electronically: October 7, 2010
MathSciNet review: 2763770
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Abstract: The width of a closed convex subset of $ n$-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension $ n \geq 3.$ In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension $ n=3$: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant and therefore are either spherical caps or pieces of tubes (canal surfaces).

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Additional Information

Henri Anciaux
Affiliation: Universidade de São Paulo, IME, Bloco A, 1010 Rua do Matão, Cidade Universitária, 05508-090 São Paulo, Brazil

Brendan Guilfoyle
Affiliation: Department of Mathematics and Computing, Institute of Technology, Tralee, County Kerry, Ireland

Received by editor(s): June 19, 2009
Received by editor(s) in revised form: August 19, 2009, August 20, 2009, and May 17, 2010
Published electronically: October 7, 2010
Additional Notes: The first author was supported by Science Foundation Ireland (Research Frontiers Program)
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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