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

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I would like to know what happens with this generalization in dimension $3$ and maybe higher. Obviously 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 inpired by this one: "Local minimum from directional derivatives in the space of convex bodies."

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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. –  Yoav Kallus Jan 26 '12 at 18:10
@Yoav, why don't you write it as an answer? –  Anton Petrunin Jan 26 '12 at 18:57
Sorry, I'm still getting used to how things work here on MO. Thanks for the MOtiquette pointer. –  Yoav Kallus Jan 26 '12 at 19:30

2 Answers 2

up vote 28 down vote accepted

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 title "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 open (e.g. a cone), and this case appears in the more complete list in Groemer's book.

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This adds nothing to Yoav Kallus' answer, but I was curious to see what these rotors look like (Schneider's notes has no figures). I found grainy photos of rotors for the cube, the regular octahedron, and the regular tetrahedron in a 50-year old paper by Michael Goldberg, "Rotors in Polygons and Polyhedra," Mathematics of Computation, Vol. 14, No. 71 (July, 1960), pp. 229-239:
(They remind me of stones found on a beach!)

Of course one can find much better examples of cube rotors, which as Anton points out, are just constant-width bodies. E.g., this is from the cover of Bryant and Sangwin's 2008 How Round Is Your Circle:
    Constant width bodies

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