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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10588-9

Published electronically:
October 7, 2010

MathSciNet review:
2763770

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Abstract | References | Similar Articles | Additional Information

Abstract: The width of a closed convex subset of -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 In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension : 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

Email:
brendan.guilfoyle@ittralee.ie

DOI:
https://doi.org/10.1090/S0002-9939-2010-10588-9

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.