On the partitionability of point sets in space (preliminary report)

We consider the problem of partitioning sets of <italic>n</italic> points in <italic>d</italic> dimensions by means of <italic>&kgr;</italic> intersecting hyperplanes. We collect known results on this problem and give some new results. In particular, for <italic>d</italic>=<italic>&kgr;</italic>=3 it is known that a set in general position can be split into equal parts given any initial bisecting plane and two other carefully chosen planes. We show that this result does not extend to the case <italic>d</italic>=<italic>&kgr;</italic>=4. We also give bounds on the smallest integer <italic>h</italic>(<italic>&kgr;</italic>) such that sets in <italic>h</italic>(<italic>&kgr;</italic>)-space can be partitioned by <italic>&kgr;</italic> hyperplanes into 2<supscrpt><italic>&kgr;</italic></supscrpt> subsets of equal cardinality, partially answering a question raised by Paul Erdös.