On empty convex polygons in a planar point set

Abstract

Let P be a set of n points in general position in the plane. Let X k (P ) denote the number of empty convex k-gons determined by P . We derive, using elementary proof techniques, several equalities and inequalities involving the quantities X k (P ) and several related quantities. Most of these equalities and inequalities are new, except for a few that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, and algebraic topology. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.

A Note on the Number of Empty Triangles

“…The lower bound on the number of empty triangles was improved by Dehnhardt [4] in his doctoral dissertation to n 2 − 5n + 10 (if n ≥ 12). However, this result in the thesis of Dehnhardt (in German) has been practically unnoticed in the community of Computational Geometry, and in fact it is not cited even in surveys on this subject like [1], [6] or [7]. In this note, we slightly improve the lower bounds given in [3] and [4] proving that any set of points P contains at least n 2 − 5n + H + 4 + 3 · n − 4 8 empty triangles, where H is the number of points on the boundary of the convex hull of P .…”

“…Probably more interesting than this new lower bound is the result that there is a well defined family of empty triangles, the family of "triangles not generated by an empty convex pentagon", containing an invariant number of empty triangles, exactly n 2 − 5n + 4 + H triangles. In this paper we are going to use the definitions and notation used in Pinchasi et al [7]. Let P = {p 1 , .…”

Lower bounds for the number of small convex k -holes

“…For consistency with previous publications, we use the same definitions and notation as in [17,11,12]. We further will recall (and slightly adapt) some statements and proofs from [12] to keep our paper self-contained.…”

On Counting and Analyzing Empty Pseudo-triangles in a Point Set

“…The results obtained in this respect are surveyed in [2]. Certain bounds on the number of convex empty polygons in a point set can be found in [6,7,11].…”