## 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).## References

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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