Shrinking the Jung radius: Maximizing partial coverage of finite point sets
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915647340937216 |
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| author | Bezdek, András Henderschedt, Owen |
| author_facet | Bezdek, András Henderschedt, Owen |
| contents | 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 $N_n(r)$ ($0 < r < \frac 1{\sqrt3}$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shrinking the Jung radius: Maximizing partial coverage of finite point sets Bezdek, András Henderschedt, Owen Combinatorics Metric Geometry 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 $N_n(r)$ ($0 < r < \frac 1{\sqrt3}$). |
| title | Shrinking the Jung radius: Maximizing partial coverage of finite point sets |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2407.03553 |