#### 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

#### 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.

## 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.