4-Holes in point sets

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 ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft

On 5-Gons and 5-Holes

“…Similar results for 4-holes can be found in [5]. For 4-gons there is a one-to-one relation to the rectilinear crossing number of the complete graph, and thus results can be found in the respective literature (e.g.…”

“…In a forthcoming paper (by Aichholzer, Fabila-Monroy, Hackl, Huemer, Pilz, and Vogtenhuber) this bound is further improved to h 5 …”

Empty non-convex and convex four-gons in random point sets

“…In [1] it was shown that every set of n points in general position in the plane determines at least 5n 2 2 − Θ(n) empty four-gons and a point set with only O(n 5/2 log n) empty four-gons is given. Our result improves the latter bound to O(n 2 log n).…”

“…Research mainly focussed on empty convex polygons. Only recently the number of convex and non-convex polygons in point sets has been studied [1,3,4].…”

The number of empty four-gons in random point sets

“…Research mainly focussed on empty convex polygons. Only recently the number of convex and non-covex polygons in point sets has been studied [1,3,4]. In [1] it is shown that every set of n points in general position in the plane determines at least 5n 2 2 − Θ(n) empty four-gons and a point set with only O(n 5/2 log n) empty four-gons is given.…”