Tight Bounds on the Expected Number of Holes in Random Point Sets

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10436870" target="_blank" >RIV/00216208:11320/21:10436870 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/978-3-030-83823-2_64" target="_blank" >https://doi.org/10.1007/978-3-030-83823-2_64</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/978-3-030-83823-2_64" target="_blank" >10.1007/978-3-030-83823-2_64</a>

Result language
angličtina
Original language name
Tight Bounds on the Expected Number of Holes in Random Point Sets
Original language description
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) and we show that our new bound is tight for $d leq 3$. 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).
Czech name
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Czech description
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Project
<a href="/en/project/GA18-19158S" target="_blank" >GA18-19158S: Algorithmic, structural and complexity aspects of geometric and other configurations</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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