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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.


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).

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