Non-Partitionable Point Sets

Let S be a finite set of n points in d-dimensional space. S is e.(n)-partitionable if there exists a set of d mutually intersecting hyperplanes that divide d-space into 2d open regions, no 2d 1 of which together contain more than o(n) points of S. Willard (1982) has shown that every set in 2-space is :n-partitionable. Yao (1983) has shown that every set in 3-space is gn-partitionable. It is shown here that there exist sets S of arbitrary cardinality in d-space, d 2 5, for which d2 +l open regions together contain at least n d2 points of S, in any partition by d intersecting hyperplanes. Further, at least 2d dZ 1 open regions contain no points of S. This implies that the powerful balanced quad and act-trees introduced by the above authors may not be generalized to balanced 2d-trees in dimension at least 5.