Disjoint Empty Convex Pentagons in Planar Point Sets

Abstract

Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of $n$ points in the plane is least $\lfloor\frac{n}{10}\rfloor$. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjoint empty convex pentagons. We also show that any set of $2m+9$ points in the plane, where $m$ is a positive integer, can be subdivided into three disjoint convex regions, two of which contains $m$ points each, and another contains a set of 9 points containing an empty convex pentagon. Combining these two results, we obtain non-trivial lower bounds on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of $n$ points in the plane, no three on a line, is at least $\lfloor\frac{5n}{47}\rfloor$. This bound has been further improved to $\frac{3n-1}{28}$ for infinitely many $n$.