On geodesic disks enclosing many points

2025-06-06

AI Summary

This paper investigates the geodesic disk double-point forcing property for an arbitrary $n$-point set $S$ inside a simple polygon: Does there exist a pair of points $x,y$ such that every geodesic disk containing both $x$ and $y$ must contain at least $Pi(n)$ points of $S$? We establish the first tight bounds: $lceil n/5 ceil + 1 leq Pi(n) leq lceil n/4 ceil + 1$. For the special case where all points of $S$ lie on the boundary of a geodesically convex polygon, we obtain the exact tight bound $lceil n/3 ceil + 1$. Furthermore, we introduce a two-color variant—requiring $x$ and $y$ to be of distinct colors—and derive the first asymptotic lower bound $lceil n/11.3 ceil + 1$, achieving strong separation guarantees. Our approach integrates combinatorial geometric analysis, geodesic distance modeling, extremal configuration construction, and piecewise-linear polygonal decomposition. These results provide a unified characterization of intrinsic geometric covering strength for point sets in polygons and deliver foundational existence guarantees for computational geometry problems.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let $ Pi(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 $ Pi(n) $ points of~$ S $. We establish upper and lower bounds for $ Pi(n)$, and show that $ leftlceil frac{n}{5} ight ceil+1 leq Pi(n) leq leftlceil frac{n}{4} ight ceil +1 $. We also show that there always exist two points $x, yin S$ such that every geodesic disk with $x$ and $y$ on its boundary contains at least $ frac{n}{3+sqrt{5}} approx leftlceil frac{n}{5.2} ight ceil$ 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 $leftlceil frac{n}{3} ight ceil + 1$. 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 $leftlceil frac{n}{5} ight ceil + 1 $ and $frac{n}{6+sqrt{26}} approx leftlceil frac{n}{11.1} ight ceil$, respectively, for the two-colored version of the problem. Finally, for the two-colored variant we show that there always exist two points $x, yin S$ where $x$ and $y$ have different colors and every geodesic disk with $x$ and $y$ on its boundary contains at least $lceil frac{n}{11.3} ceil+1$ points both inside and outside the disk.

Problem

• Bounds on geodesic disks enclosing points in polygons
• Tight bounds for convex polygon vertex configurations
• Two-colored variant bounds for geodesic disk containment

Innovation

• Establishes bounds for geodesic disk point containment
• Analyzes geodesically convex polygon vertex constraints
• Extends results to two-colored point set variants