On the number of empty convex quadrilaterals of a finite set in the plane

Abstract

Let P be a set of n points in the plane, no three collinear. A convex polygon of P is called empty if no point of P lies in its interior. An empty partition of P is a partition of P into empty convex polygons. Let k be a positive integer and N π k (P) be the number of empty convex k-gons in an empty partition π of P. Define g k (P) =: max{N π k (P) : π is an empty partition of P}, G k (n) =: min{g k (P) : |P| = n}. We mainly study the case of k = 4 and get the result that G 4 (n) ≥ 9n 38 . For specified n = 21 × 2 k−1 − 4 (k ≥ 1), we obtain the better bound G 4 (n) ≥ 5n−1 21 .

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

“…The remaining case of k = 6 was recently solved independently by Gerken [18] and Nicolás [29], who proved that every sufficiently large set of points in general position contains a 6-hole. See [3,4,7,8,10,22,24,25,30,31,32,35,36,37,38,39,40,41] for more on empty convex polygons.…”

A bound on a convexity measure for point sets

“…Urabe and Hosono [9,4] studied the fewest number of parts in a partition of a point set into vertex sets of disjoint convex polygons; this was previously explored by Chazelle and Dobkin [2] for fixed polygons rather than point sets. In a similar vein, Wu and Ding [11] studied for point sets the maximum number of subsets in convex position that, though not necessarily disjoint, have no other points in their interior.…”

Compatible 4-Holes in Point Sets

“…Hosono and Urabe [19] proved that the number of disjoint 4-holes is at least 5n/22 ; they improved this bound to (3n−1)/13 when n = 13 • 2 k −4 for some k 0. A variant of this problem where the 4-holes are vertex-disjoint, but can overlap, is considered in [29]. As for compatible holes, it is easy to verify that the number of compatible 3-holes in any n-set is at least n−2 and at most 2n−5; these bounds are obtained by triangulating the point set: we get n−2 triangles, when the point set is in convex position, and 2n−5 triangles, when the convex hull of the point set is a triangle.…”