Partitioning point sets in space into disjoint convex polytopes

Abstract We study the problem of partitioning point sets in the space so that each equivalence class is a convex polytope disjoint from the others. For a set of n points P in R 3 , define f(P) to be the minimum number of sets in a partition into disjoint convex polytopes of P and F(n) as the maximum of f(P), over all sets P of n points. We show that ⌈n/2(log2n+1)⌉≤F(n)≤⌈2n/9⌉. The lower bound also holds for partition into empty convex polytopes.