On the three-dimensional Blaschke-Lebesgue problem

Henri Anciaux; Brendan Guilfoyle

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arXiv · 0906.3217

On the three-dimensional Blaschke-Lebesgue problem

Henri Anciaux·Brendan Guilfoyle

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Abstract

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. 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 > 2. 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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Henri Anciaux, Brendan Guilfoyle. 2009-06-17. On the three-dimensional Blaschke-Lebesgue problem. https://arxiv.org/abs/0906.3217

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