Separating translates in the plane: Combinatorial bounds and an algorithm

Abstract

In this paper, we establish two combinatorial bounds related to the separation problem for sets of n pairwise disjoint translates of convex objects: 1) there exists a line which separates one translate from at least (Formula presented) translates, for some constant c that depends on the “shape” of the translates and 2) there is a function f such that there exists a line with orientation θ or f(θ) which separates one translate from at least (Formula presented) - 4 translates, for any orientation θ (f is defined only by the “shape” of the translate). We also present an O(n log(n + k) + k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst sets of n pairwise disjoint translates of convex k-gons.