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."
 A: 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.
A: 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 (Wayback Machine and an alternative link):
   


