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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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  • [AG] H. Anciaux, N. Georgiou, The Blaschke-Lebesgue problem for constant width bodies of revolution, arXiv:0903.4284.
  • [Ba] T. Bayen, PhD Dissertation, Optimisation de formes dans la classe des corps de largeur constante et des rotors, Université Pierre et Marie Curie - Paris 6, 2007.
  • [BLO] T. Bayen, T. Lachand-Robert, E. Oudet, Analytic parametrizations and volume mimization of three dimensional bodies of constant width, Arch. Ration. Mech. Anal. 186 (2007), no. 2, 225-249. MR 2342202 (2008f:52005)
  • [CCG] S. Campi, A. Colesanti, P. Gronchi, Minimum problems for volumes of convex bodies, in Partial differential equations and applications, Vol. 177 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1996, 43-55. MR 1371579 (96j:52012)
  • [CG] G.D. Chakerian, H. Groemer, Convex bodies of constant width, in Convexity and its applications (Ed. P. Gruber and J. Wills), Birkhäuser, Basel, 1983, 49-96. MR 731106 (85f:52001)
  • [GK] B. Guilfoyle, W. Klingenberg On $ C^2$-smooth surfaces of constant width, Tbilisi Math. Journal 1 (2009), 1-17. MR 2574869
  • [GW] P.-M. Gruber, J.-M. Wills, Handbook of convex geometry, North-Holland, Amsterdam, 1993. MR 1242973 (94e:52001)
  • [Ha] E. Harrell, II, A direct proof of a theorem of Blaschke and Lebesgue, J. Geom. Anal. 12 (2002), no. 1, 81-88. MR 1881292 (2002k:52009)
  • [Ho] R. Howard, Convex bodies of constant width and constant brightness, Adv. Math. 204 (2006), no. 1, 241-261. MR 2233133 (2007f:52004)
  • [ST] K. Shiohama, R. Takagi, A characterization of a standard torus in $ E^3,$ J. of Diff. Geom. 4 (1970), 477-485. MR 0276906 (43:2646)

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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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