4-Holes in point sets Article uri icon

abstract

  • We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n≥9, the maximum number of general 4-holes is (n4); the minimum number of general 4-holes is at least 5/2n 2-Θ(n); and the maximum number of non-convex 4-holes is at least 1/2n3-Θ(n2logn) and at most 1/2n 3-Θ(n2). © 2014 Elsevier B.V.

publication date

  • 2014-01-01