On some probabilistic inequalities in Banach spaces

The following question was brought to my attention by Assaf Naor:
for a given separable Banach space (F,||*||), what is the smallest
constant K in the inequlity below such that

inf_{z in F} (E||X-z||+E||Y-z||) \leq K E||X-Y||

for any pair of independent integrable F-valued random vectors X and Y. The answer depends on the geometry of F. We will discuss this and some
related problems.