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&mdash;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