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- W4280560128 abstract "For a finite set $Asubset mathbb{R}^d$, let $Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $gamma_d(n)$ denote the largest integer such that any set $A$ of $n$ points in general position in $mathbb{R}^d$, satisfying $Delta(A) leq alpha n^{1/d}$ for a fixed $alpha>0$, contains at least $gamma_d(n)$ points in convex position. About $30$ years ago, Valtr proved that $gamma_2(n)=Theta(n^{1/3})$. Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensions and prove that $gamma_3(n) =Theta(n^{1/2})$. The lower bound implies the following approximation: Given any $n$-element point set $Asubset mathbb{R}^3$ in general position, satisfying $Delta(A) leq alpha n^{1/3}$ for a fixed $alpha$, a $Omega(n^{-1/6})$-factor approximation of the maximum-size convex subset of points can be computed by a randomized algorithm in $O(n log{n})$ expected time." @default.
- W4280560128 created "2022-05-22" @default.
- W4280560128 creator A5030799674 @default.
- W4280560128 creator A5091313409 @default.
- W4280560128 date "2022-05-06" @default.
- W4280560128 modified "2023-10-17" @default.
- W4280560128 title "Finding Points in Convex Position in Density-Restricted Sets" @default.
- W4280560128 doi "https://doi.org/10.48550/arxiv.2205.03437" @default.
- W4280560128 hasPublicationYear "2022" @default.
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