In the case of the cube, it was shown by Bourgain (very recently) 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 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 Kali's blog here).

In the case of the cube, it was shown by Bourgain (very recently) 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 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).

In the case of the cube, it was shown by Bourgain (very recently) that the constants remain independent of $n$ for all $p>1$. It seems to remain open for 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."

In the case of the cube, it was shown by Bourgain (very recently) 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 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 Kali's blog here).