The Blaschke-Lebesgue problem for constant width bodies of revolution

We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry. 2000 MSC: 52A15 Introduction The width of a convex body B in n-dimensional Euclidean space in the direction ~u is the distance between the two supporting planes of B which are orthogonal to ~u. When this distance is independent of ~u, B is said to have constant width. The ratio I(B) of the volume of a constant width body to the volume of the ball of the same width is homothety invariant, as is the isoperimetric ratio. Moreover the maximum of I(B) is attained by round spheres, just as the minimum of the isoperimetric ratio. However, while the latter is not bounded from above, the infimum of I is strictly positive, since compactness properties of the space of convex sets ensures the existence of a minimizer. It is known since the works of Blaschke and Lebesgue that the Reuleaux triangle, obtained by taking the intersection of three discs centered at the vertices of an equilateral triangle, minimizes I in dimension n = 2. The determination of the minimizer of I in any dimension is the BlaschkeLebesgue problem. Recently several simpler solutions of the problem in dimension 2 have been given (cf [Ba],[Ha]), however the Blaschke-Lebesgue problem in dimension n = 3 appears to be very difficult to solve and remains open. In this paper we prove: ∗The first author is supported by SFI (Research Frontiers Program)