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On geodesic disks enclosing many points

Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo Silveira, Tyler Tuttle

Abstract

Let $ Π(n) $ be the largest number such that for every set $ S $ of $ n $ points in a polygon~$ P $, there always exist two points $ x, y \in S $, where every geodesic disk containing $ x $ and $ y $ contains $ Π(n) $ points of~$ S $. We establish upper and lower bounds for $ Π(n)$, and show that $ \left\lceil \frac{n}{5}\right\rceil+1 \leq Π(n) \leq \left\lceil \frac{n}{4} \right\rceil +1 $. We also show that there always exist two points $x, y\in S$ such that every geodesic disk with $x$ and $y$ on its boundary contains at least $ \frac{n}{7+\sqrt{37}} \approx \left\lceil \frac{n}{13.1} \right\rceil$ points both inside and outside the disk. For the special case where the points of $ S $ are restricted to be the vertices of a geodesically convex polygon we give a tight bound of $\left\lceil \frac{n}{3} \right\rceil + 1$. We provide the same tight bound when we only consider geodesic disks having $ x $ and $ y $ as diametral endpoints. We give upper and lower bounds of $\left\lceil \frac{n}{5} \right\rceil + 1 $ and $\frac{n}{6+\sqrt{26}} \approx \left\lceil \frac{n}{11.1} \right\rceil$, respectively, for the two-colored version of the problem. Finally, for the two-colored variant we show that there always exist two points $x, y\in S$ where $x$ and $y$ have different colors and every geodesic disk with $x$ and $y$ on its boundary contains at least $\left\lceil \frac{n}{27.1}\right\rceil+1$ points both inside and outside the disk.

On geodesic disks enclosing many points

Abstract

Let be the largest number such that for every set of points in a polygon~, there always exist two points , where every geodesic disk containing and contains points of~. We establish upper and lower bounds for , and show that . We also show that there always exist two points such that every geodesic disk with and on its boundary contains at least points both inside and outside the disk. For the special case where the points of are restricted to be the vertices of a geodesically convex polygon we give a tight bound of . We provide the same tight bound when we only consider geodesic disks having and as diametral endpoints. We give upper and lower bounds of and , respectively, for the two-colored version of the problem. Finally, for the two-colored variant we show that there always exist two points where and have different colors and every geodesic disk with and on its boundary contains at least points both inside and outside the disk.

Paper Structure

This paper contains 15 sections, 21 theorems, 18 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Euclidean and geodesic disks: some similarities and differences
  4. Some similarities
  5. Some differences
  6. The general case
  7. A Constant Fraction of Points Inside Geodesic Disks
  8. A Constant Fraction of Points Inside and Outside Geodesic Disks
  9. Sets in geodesically convex position
  10. Diametral geodesic disk
  11. Bichromatic geodesic disks
  12. A Constant Fraction of Bichromatic Points Inside Geodesic Disks
  13. A Constant Fraction of Bichromatic Points Inside and Outside Geodesic Disks
  14. Upper bound
  15. Conclusion

Key Result

Lemma 1

Let $p, q$ be two points in $P$ and $D, D'$ be two geodesic disks through $p$ and $q$, respectively with centers $c$ and $c'$. If $c'$ is left of $\ell(p,c)$ then $D\cap P^+(p,q)\subset D'$.

Figures (8)

  • Figure 1: Geodesic disk centered at $c$, with radius $\lvert g(c,z)\rvert$, shown solid, and the equivalent Euclidean disk, shown dashed.
  • Figure 2: The point $a$ belongs to the interior of $\triangle (c',p,q)$.
  • Figure 3: The intersection of the disk through $p$ and $q$ with center at $c$ (in red) and $P^+(p,q)$ (in gray) is contained in the disk through $p$ and $q$ with center at $c'$ (in blue).
  • Figure 4: Any geodesic disk through $u$ and $v$ contains at least one endpoint of $g(p,q)$.
  • Figure 5: The angle $\angle uvw$ is greater than $\frac{\pi}{3}$, and $\lvert vq\rvert < \lvert vp\rvert$.
  • ...and 3 more figures

Theorems & Definitions (44)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 34 more