Shrinking the Jung radius: Maximizing partial coverage of finite point sets

TL;DR

This paper investigates a fractional version of Jung's theorem for finite point sets in the plane, determining exact maximum coverage numbers for specific disk radii and providing bounds for others.

Contribution

It establishes exact values of maximum point coverage by disks of radius 1/2 and 1/4 for finite point sets, extending Jung's theorem to fractional coverage scenarios.

Findings

For radius 1/2, maximum coverage is approximately one-third of points plus one.

For radius 1/4, maximum coverage depends on divisibility of n by 7.

Provides bounds for coverage for radii between 0 and 1/√3.

Abstract

Jung's theorem says that planar sets of diameter $1$ can be covered by a closed circular disk of radius $\frac 1{\sqrt3}$. In this paper we consider a fractional Jung-type problem for finite planar point-sets. Let $\mathcal{P}_n$ be the family of all finite sets of $n$ points in the plane, of diameter at most $1$. Let the function value $N_n(r)$ ($0 < r \leq 1$) be the largest integer $k$ so that for every point set $P \in \mathcal{P}_n$ there is a closed circular disk of radius $r$ which covers at least $k$ points of $P$. We focus on the radii $r=\frac 12$ and $r=\frac 14$ and prove exact maximum values. Concerning the radius $r= \frac 12$, we prove $N_n(\frac{1}{2})=\lceil \frac{n}{3}\rceil+1$. Concerning the radius $r= \frac 14$, we prove that $N_{n}(\frac{1}{4}) = \lceil \frac{n}{7}\rceil$ if $n$ is not a multiple of 7, and $N_{n}(\frac{1}{4})$ is $ \frac{n}{7}$ or $ \frac{n}{7}+1$ otherwise. We also initiate further study of the function $N_n(r)$ by giving lower and upper bounds for $N_n(r)$ ($0 < r < \frac 1{\sqrt3}$).