This relevant paper was just released:

> Howard Resnikoff. "On Curves and Surfaces of Constant Width." Apr. 2015.
([arXiv abstract](https://arxiv.org/abs/1504.06733).)

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

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![ConstWidth][1]
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  [1]: https://i.sstatic.net/Az8Ef.png