There are many 5-holes

Abstract

Given a set P of points on the plane, a polygon with vertices in P is said to be empty if it contains no element of P in its interior. We show that every set of n points in general position on the plane determines at least $Ω(n^{20/11})$ empty convex pentagons (also known as 5-holes). This result improves upon the previous bound of $Ω(n\cdot(\log n)^{4/5})$ obtained by Aicholzer et al. [JCT A, 2020], and significantly narrows the gap with respect to the conjectured $Ω(n^2)$ lower bound (which, if true, would be tight). Unlike some of the other works in this line of research, our proof does not require computer assistance.

Figures (11)

• Figure 1: The image shows two possible configurations of three distinct points $a$, $b$ and $c$, along with the angle $\angle abc$ determined by them. On the left, we have that $\measuredangle abc<\pi$, while on the right $\measuredangle abc>\pi$.
• Figure 2: The picture shows a set $P$ of $13$ points with $L(P)=3$. The outer-most layer is $\ell_1$, the inner-most layer is $\ell_3$, and the middle layer is $\ell_2$.
• Figure 3: The picture shows a set of $7$ points on the plane. The point $p$ lies on the boundary of its convex hull, and the other $6$ points have been labeled according to the radial order with respect to $p$.
• Figure 4: On the left, a portion of the layer $\ell_i$ has been highlighted in blue. In particular, we see the point $p$ and its adjacent vertices $q$ and $r$. The triangle $\triangle pab$ is empty, and we assume that the line $\overleftrightarrow{ab}$ crosses neither $\overline{pq}$ nor $\overline{pr}$. Then, $pqabr$ is a convex pentagon. On the right, we see the points $p'$ and $r'$, which have been chosen as follows. Out of the elements of $P$ contained in $\triangle pqa$ (other than $p$ and $a$), $q'$ is the one closest to $\overleftrightarrow{pa}$. Out of the elements of $P$ contained in $\triangle pbr$ (other than $p$ and $b$), $r'$ is the one closest to $\overleftrightarrow{pb}$. The points $p$, $q'$, $a$, $b$ and $r'$ are the vertices of a $5$-hole (highlighted in red).
• Figure 5: On the left, we see the convex hull $R$ of $P_{ab}$. If $p\in P$ is a point in the interior of $R$, then the segments $\overline{pa}$ and $\overline{pb}$ must cross the same side of $R$. It follows that $pqabr$ is a convex pentagon. As in the proof of Theorem \ref{['thm:layers']}, there must exist some $5$-hole of the form $pq'abr'$.

Theorems & Definitions (23)

• Theorem 1.1: Aicholzer et al. aichholzer2020superlinear
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:layers']}
• Lemma 3.1
• proof
• Theorem 3.2
• proof