The standard Reuleaux triangle is not smooth, but the three points of tangential discontinuity can be smoothed as in the figure below (left), from [the Wikipedia article][1]. However, it is unclear (to me) from this diagram whether the curve is $C^2$ or $C^\infty$. _Meissner’s tetrahedron_ is a 3D body of constant width, but it is not smooth, as is evident in the right figure below. <br /> ![Constant Width][2] <br /> My question is: > Are there $C^\infty$ constant-width bodies in $\mathbb{R}^d$ (other than the spheres)? The image of Meissner’s tetrahedron above is taken from [the impressive work][3] of Thomas Lachand–Robert and Edouard Oudet, "Bodies of constant width in arbitrary dimension" (_Math. Nachr._ 280, No. 7, 740-750 (2007); [pre-publication PDF here][4]). I suspect the answer to my question is known, in which case a reference would suffice. Thanks! <b>Addendum.</b> Thanks to the knowledgeable (and rapid!) answers by Gerry, Anton, and Andrey, my question is completely answered—I am grateful!! [1]: http://en.wikipedia.org/wiki/Curve_of_constant_width [2]: http://cs.smith.edu/~orourke/MathOverflow/ConstantWidth.jpg [3]: http://www.lama.univ-savoie.fr/~lachand/Spheroforms.html [4]: http://www.lama.univ-savoie.fr/~lachand/pdfs/spheroforms.pdf