# Bodies of constant width?

In two-dimensional case one can generalize figures of constant width as figures which can rotate in a convex polygon. Here is one example which can be used to drill triangular holes:

I would like to know what happens with this generalization in dimension $$3$$ and maybe higher. Obviously a body of constant width $$1$$ can rotate arbitrary in a unit cube. More formally, given a body $$B$$ of constant width $$1$$ and $$A\in SO(3)$$ there is $$v\in \mathbb R^3$$ such that $$A(B)+v\subset\square,$$ where $$\square$$ is unit cube. On the other hand, except for the cube, I do not see any other examples of convex polyhedron which have nontrivial rotating bodies (i.e. distinct from the inscribed ball).

I hope that the answer is known. (= I hope I should wait for the answer and I do not have to think.)

The question is inspired by this one: "Local minimum from directional derivatives in the space of convex bodies."

• These bodies are called "rotors" of the corresponding cavity. A full classification of non-trivial (i.e. other than bodies of constant width) rotors is known. A good reference on these is "Geometric Applications of Fourier Series and Spherical Harmonics" by Helmut Groemer. As I recall, only the regular simplex and cross-polytope admit non-trivial rotors in dimensions four and above. In three-dimensions, there are more cases. Jan 26, 2012 at 18:10
• @Yoav, why don't you write it as an answer? Jan 26, 2012 at 18:57
• Sorry, I'm still getting used to how things work here on MO. Thanks for the MOtiquette pointer. Jan 26, 2012 at 19:30

I found the reference I was looking for. The full list of cases under which $$K$$ is a rotor in a cavity shaped like the polytope $$P$$ is available on page 27 of the notes titled "The use of spherical harmonics in convex geometry" by Rolf Schneider. They are available under "Course Materials" on his website. As I recall, there is one more non-trivial case in $$d=3$$ if the cavity is allowed to be unbounded (e.g. a cone), and this case appears in the more complete list in Groemer's book.