In the case of the cube, it was shown by Bourgain ([very recently][1]) that the constants remain independent of $n$ for all $p>1$. The problem appears to be open for the case of more general centrally symmetric convex bodies. On page 3 of his recent preprint, Bourgain writes:

"While it is reasonable to believe that this statement holds in general, our argument is based on a very explicit analysis which does not immediately carry over to other convex symmetric bodies."

For $p=1$, J. Aldaz [has shown][2] that the (weak $L^1$) constant can't be taken independent of $n$ in the case of a cube.  The case of the ball is open (this problem was briefly discussed on Gil Kalai's blog [here][3]).


  [1]: http://arxiv.org/abs/1212.2661
  [2]: http://arxiv.org/abs/0805.1565
  [3]: http://gilkalai.wordpress.com/2012/11/17/a-few-mathematical-snapshots-from-india-icm2010/