On the three-dimensional Blaschke-Lebesgue problem

Anciaux, Henri; Guilfoyle, Brendan

doi:10.1090/S0002-9939-2010-10588-9

On the three-dimensional Blaschke-Lebesgue problem
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by Henri Anciaux and Brendan Guilfoyle

Proc. Amer. Math. Soc. 139 (2011), 1831-1839

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

Published electronically: October 7, 2010

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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).

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