Locked convex polyhedra Call a set of polyhedra free if it is possible to rigidly move the polyhedra, without any polyhedron intersecting any other, so that their pairwise distances are arbitrary large, and locked otherwise. So two linked tori are locked, as is a ship in a bottle.
Can a finite set of convex polyhedra in $\mathbb{R}^3$ ever be locked?
note: We can move these polyhedra simultaneously. 
 A: I have already up-voted Allen's answer (see above). His solution is discrete though. I thought I would write a comment but it's simpler to provide an extension of his method right here. I will even generalize his approach slightly. You may up-vote Allen for all this, instead of my answer.
Let's fix a selection function--given an arbitrary convex body $B\subseteq\mathbb R^n$, let $s(B)\in B$.
Let $F$ be a family of pairwise disjoint convex bodies in $\mathbb R^n\,$ (bodies $\,$means$\,$ compact and n-dimensional). $\,$Define
$$\forall_{t\ge 1}\ T_t(B)\ := \{x\in\mathbb R^n\,:\, x-(t+1)\cdot s(B)\in B\}$$
Then $T_t\ $ (for $t\ge 1)\ $ move convex bodies $B\in F$ rigidly, and keep them separated all the time, where the separation goes to $\infty$ when $t\rightarrow \infty$.

In particular, one can select $s(B)$ to be the center of gravity of $B$.

REMARK $\,$ We may decide on $s(B)$ being the center of gravity of $B$, as Allen has done. Then $\ \forall_{t\ u\ge 1}\ T_{t\cdot u} = T_u\circ T_t$.  
A: I also asked this question in M.SE and at there Igor Pak gives a quite useful reference book which solve this problem completely. 
A: No locking, even if you restrict to translations. Scale the whole arrangement up by a factor of $c$, then scale each polyhedron down by $c$ around its center of mass. Neither step introduces collisions. Now, that's not quite rigid motion like you asked for, so instead do this recipe $N$ times with the factor $\sqrt[N]{c}$, and let $N\to \infty$. 
A: To supplement your references, this class of problems has been studied
since the 1980's:

Toussaint, Godfried. "Movable separability of sets." Computational Geometry. 1985.


          


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A: you can use tetrahedrons to cover a sphere in such a way that the two high points of each tetrahedron cover low points of other tetrahedrons, as if they were in a square lattice. not sure what to do where the lattice breaks though. this kind of stracture should stop them from moving one at a time (allen already solved the case otherwise).
