On the Three-Dimensional Blaschke-Lebesgue Problem

Authors

Henri Anciaux, Universidade De Sao Paulo, Ime, Bloco A, 1010 Rua Do Mat ˜ Ao, Cidade Universit ˜ Aria, ´05508-090 Sao Paulo, Brazil.
Brendan Guilfoyle, Department of Mathematics and Computing, Institute of Technology, Tralee, County Kerry, IrelandFollow

Document Type

Article

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Disciplines

Mathematics

Publication Details

Proceedings of the American Mathematical Society

© Copyright 2010 American Mathematical Society.

First published in Proc. Amer. Math. Soc. 139 (May 2011), published by the American Mathematical Society. © 2016 American Mathematical Society.

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 ≥ 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).

Recommended Citation

Anciaux, H. & Guilfoyle, B., 2011. On the three-dimensional Blaschke-Lebesgue problem. Proceedings of the American Mathematical Society, 139(05), pp.1831–1831. Available at: http://dx.doi.org/10.1090/S0002-9939-2010-10588-9.