Planar point sets with a small number of empty convex polygons

Abstract

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

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

“…Concerning empty convex four-gons, Bárány and Füredi [7] established that f 4 (n) is of order Θ(n 2 ), and the currently best bounds on f 4 (n) are [2,9]. Research mainly focussed on empty convex polygons.…”

On empty convex polygons in a planar point set

“…In general, the lower bounds cannot be super-quadratic, as has been noted in several papers [5,8]. The construction with the best upper bounds is due to Bárány and Valtr [5]; it produces n-point sets with roughly 1.62n 2 empty triangles, 1.94n 2 empty convex quadrilaterals, 1.02n 2 empty convex pentagons, and 0.2n 2 empty convex hexagons.…”

“…The construction with the best upper bounds is due to Bárány and Valtr [5]; it produces n-point sets with roughly 1.62n 2 empty triangles, 1.94n 2 empty convex quadrilaterals, 1.02n 2 empty convex pentagons, and 0.2n 2 empty convex hexagons. Both constructions in [5,8] use Horton's construction as the main building block.…”

The number of empty four-gons in random point sets

“…For the case of empty triangles, Katchalski and Meir [11] showed that f 3 (n) is of order Θ(n 2 ). Later, this bound has been refined [2,6,7,8,9,13]; the currently best bounds are n 2 − 32n…”

“…Concering empty convex four-gons, Bárány and Füredi [6] established that f 4 (n) is of order Θ(n 2 ), and the currently best bounds on [2,7]. Research mainly focussed on empty convex polygons.…”