Connectedness of parts used in the Banach–Tarski paradox 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).

Dekker and de Groot proved (Decompositions of a sphere,  Fund. Math. 43 (1956), 185–194) that the pieces can be made to be totally imperfect, and hence connected and locally connected.  I don't know about path connected, though.
Although you didn't ask about this, one can also argue that "decompose and reassemble" implicitly suggests that the rearrangement can take place using continuous motions during which the pieces never intersect.  That this is possible was proved by Trevor Wilson
(A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's problem, J. Symb. Logic 70 (2005), 946–952).  But I don't know if there is a common generalization of the Dekker–de Groot result above and Wilson's result.
