TITLE: Are there smooth bodies of constant width?
QUESTION [21 upvotes]: 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.
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.
   
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 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). Here is a link to Wayback Machine.)
I suspect the answer to my question is known, in which case a reference would suffice.  Thanks!
Addendum.
Thanks to the knowledgeable (and rapid!) answers by Gerry, Anton, and Andrey,
my question is completely answered—I am grateful!!

REPLY [7 votes]: This relevant paper was just released:

Howard Resnikoff. "On Curves and Surfaces of Constant Width." Apr. 2015.
(arXiv abstract.)


"...we provide a Fourier series-based construction that produces arbitrarily many new surfaces of constant width."


     
![ConstWidth][1]