The average distance property of classical Banach spaces II
  A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x_1,...,x_n in the unit sphere of X there is some x in the unit sphere of X such that   1/n \sum_{k=1}^n ||x_k-x|| = r.   We show that l_p does not have the average distance property if p>2. This completes the study of the ADP for l_p spaces. 
Aicke Hinrichs and J. Wenzel