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:

![alt text][1]

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][2]."


  [1]: http://upload.wikimedia.org/wikipedia/commons/a/a0/Dvuug.gif
  [2]: http://mathoverflow.net/questions/86653/