Maintaining the minimal distance of a point set in less than linear time

Consider a set of n points in d-dimensional space. It is shown that the ordered sequence of O(n2/3) smallest distances defined by these points can be computed in optimal O(nlog n) time and O(n) space. Here, distances are measured in an arbitrary Lₜ-metric, where 1≤ t≤∞. This result is used to give a dynamic data structure of linear size, that maintains the minimal distance of the n points in O(n2/3log n) time per update.