- (with W. Klingenberg) On Cē-smooth surfaces of constant width, Tbilisi Math. J. 2 (2009) 1-17. Preprint
An integral inequality for constant width surfaces in Euclidean 3-space is established. This is used to prove that the ratio of volume to cubed width
of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have
rational support function and construct explicit examples.
- (with H. Anciaux) On the three-dimensional Blaschke-Lebesgue problem, Proc. Amer. Math. Soc. Preprint
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 greater than 2. In this paper we describe a
necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension 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).