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

TL;DR

The paper investigates fractional Jung-type questions for finite planar point sets of diameter 1 by introducing $N_n(r)$, the maximum number of points that can be forced into a circle of radius $r$ for all $n$-point unit-diameter sets. It establishes exact values for two key radii: $N_n(\tfrac{1}{2})=\lceil n/3\rceil+1$ and $N_n(\tfrac{1}{4})=\lceil n/7\rceil$ (with special behavior when $n$ is a multiple of 7), using a blend of geometric constructions, projection arguments, and carefully designed counterexamples. The work also develops general bounds and the asymptotic function $c(r)=\lim_{n\to\infty} N_n(r)/n$ across $0<r<1/\sqrt{3}$, summarizing current knowledge in a table/diagram and outlining open problems and higher-dimensional generalizations. The results illuminate how much of a unit-diameter point set can be guaranteed to lie within a fixed-radius circle and lay groundwork for broader fractional covering problems. The methods combine universal covers, circle packings, and balanced constructions to derive tight bounds and guide future research.

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 $N_n(r)$ ($0 < r < \frac 1{\sqrt3}$).

Figures (9)

• Figure 1: Circle coverings of a regular hexagon found by Y. Liu in Y22
• Figure 2: Proof of Claim 2.
• Figure 3: Example \ref{['ex:Emplus2']}
• Figure 4: Enlarged lower part of Figure \ref{['fig:mplus2']}
• Figure 5: Example \ref{['ex:fourth']}

Theorems & Definitions (13)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Example 1
• Example 2
• Proposition 1
• Proposition 2
• Proposition 3