A quote from the Wikipedia article "Axiom of choice":

One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.

I know that at least some of the parts (called *pieces* here) must be non-measurable sets. I wonder if each of them can be chosen to be a path-connected set (otherwise it's really misleading to call them *pieces*, I think).