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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).
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.