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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the three-dimensional Blaschke-Lebesgue problem
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by Henri Anciaux and Brendan Guilfoyle PDF
Proc. Amer. Math. Soc. 139 (2011), 1831-1839 Request permission

Abstract:

The width of a closed convex subset of $n$-dimensional Euclid- ean 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
  • Email: henri.anciaux@gmail.com
  • Brendan Guilfoyle
  • Affiliation: Department of Mathematics and Computing, Institute of Technology, Tralee, County Kerry, Ireland
  • MR Author ID: 631268
  • Email: brendan.guilfoyle@ittralee.ie
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1831-1839
  • MSC (2010): Primary 52A40, 52A15
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10588-9
  • MathSciNet review: 2763770