Tight bounds on the expected number of holes in random point sets

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10453242" target="_blank" >RIV/00216208:11320/23:10453242 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=KZWct6ysrl" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=KZWct6ysrl</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/rsa.21088" target="_blank" >10.1002/rsa.21088</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Tight bounds on the expected number of holes in random point sets

Popis výsledku v původním jazyce

For integers $d geq 2$ and $k geq d+1$, a emph{$k$-hole} in a set $S$ of points in general position in $mathbb{R}^d$ is a $k$-tuple of points from $S$ in convex position such that the interior of their convex hull does not contain any point from $S$. For a convex body $K subseteq mathbb{R}^d$ of unit $d$-dimensional volume, we study the expected number $EH^K_{d,k}(n)$ of $k$-holes in a set of $n$ points drawn uniformly and independently at random from $K$.We prove an asymptotically tight lower bound on $EH^K_{d,k}(n)$ by showing that, for all fixed integers $d geq 2$ and $kgeq d+1$, the number $EH_{d,k}^K(n)$ is at least $Omega(n^d)$. For some small holes, we even determine the leading constant $lim_{n to infty}n^{-d}EH^K_{d,k}(n)$ exactly. We improve the currently best known lower bound on $lim_{n to infty}n^{-d}EH^K_{d,d+1}(n)$ by Reitzner and Temesvari~(2019). In the plane, we show that the constant $lim_{n to infty}n^{-2}EH^K_{2,k}(n)$ is independent of $K$ for every fixed $k geq 3$ and we compute it exactly for $k=4$, improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche~(2015) and by the authors~(2020).

Název v anglickém jazyce

Tight bounds on the expected number of holes in random point sets

Popis výsledku anglicky

For integers $d geq 2$ and $k geq d+1$, a emph{$k$-hole} in a set $S$ of points in general position in $mathbb{R}^d$ is a $k$-tuple of points from $S$ in convex position such that the interior of their convex hull does not contain any point from $S$. For a convex body $K subseteq mathbb{R}^d$ of unit $d$-dimensional volume, we study the expected number $EH^K_{d,k}(n)$ of $k$-holes in a set of $n$ points drawn uniformly and independently at random from $K$.We prove an asymptotically tight lower bound on $EH^K_{d,k}(n)$ by showing that, for all fixed integers $d geq 2$ and $kgeq d+1$, the number $EH_{d,k}^K(n)$ is at least $Omega(n^d)$. For some small holes, we even determine the leading constant $lim_{n to infty}n^{-d}EH^K_{d,k}(n)$ exactly. We improve the currently best known lower bound on $lim_{n to infty}n^{-d}EH^K_{d,d+1}(n)$ by Reitzner and Temesvari~(2019). In the plane, we show that the constant $lim_{n to infty}n^{-2}EH^K_{2,k}(n)$ is independent of $K$ for every fixed $k geq 3$ and we compute it exactly for $k=4$, improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche~(2015) and by the authors~(2020).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA21-32817S" target="_blank" >GA21-32817S: Algoritmické, strukturální a složitostní aspekty geometrických konfigurací</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Random Structures and Algorithms

ISSN

1042-9832

e-ISSN

—

Svazek periodika

2023

Číslo periodika v rámci svazku

62

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

23

Strana od-do

29-51

Kód UT WoS článku

000789578600001

EID výsledku v databázi Scopus

2-s2.0-85128513177