Selecting distances in arrangements of hyperplanes spanned by points

In this paper we consider a problem of distance selection in the arrangement of hyperplanes induced by n given points. Given a set of n points in d-dimensional space and a number k, 1 ≤ k ≤ (^d), determine the hyperplane that is spanned by d points and at distance ranked by k from the origin. For the planar case we present an O(n log² n) runtime algorithm using parametric search partly different from the usual approach [N. Megiddo, J. ACM 30 (1983) 852]. We establish a connection between this problem in 3-d and the well-known 3 SUM problem using an auxiliary problem of counting the number of vertices in the arrangement of n planes that lie between two sheets of a hyperboloid. We show that the 3-d problem is almost 3SUM-hard and solve it by an O(n² log² n) runtime algorithm. We generalize these results to the d-dimensional (d ≥ 4) space and consider also a problem of enumerating distances.