Distinct distances in three and higher dimensions

Improving an old result of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl, we show that the number of distinct distances determined by a set P of n points in three-dimensional space is Omega(n(77/141-epsilon)) = Omega(n(0.546)), for any epsilon > 0. Moreover, there always exists a point p is an element of P from which there are at least so many distinct distances to the remaining elements of P. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.