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SUMMARY:Natan Rubin (Ben-Gurion University)
DTSTART:20210708T141500Z
DTEND:20210708T160000Z
DTSTAMP:20260422T065957Z
UID:CJCS/23
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CJCS/23/">St
 ronger bounds for weak epsilon-nets in higher dimensions</a>\nby Natan Rub
 in (Ben-Gurion University) as part of Copenhagen-Jerusalem Combinatorics S
 eminar\n\n\nAbstract\nGiven a finite point set $P$ in $R^d$\, and $\\varep
 silon>0$ we say that a point set $N$ in  $R^d$ is a weak $\\varepsilon$-ne
 t if it pierces every convex set $K$ with $|K\\cap P|\\geq \\varepsilon |P
 |$.\n \nLet $d\\geq 3$. We show that for any finite point set in $R^d$\, a
 nd any $\\varepsilon>0$\, there exists a weak $\\varepsilon$-net of cardin
 ality $O(1/\\varepsilon^{d-1/2+\\delta})$\, where $\\delta>0$ is an arbitr
 ary small constant. \n\n\nThis is the first improvement of the bound of $O
 ^*(1/\\varepsilon^d)$ that was obtained in 1993 by Chazelle\, Edelsbrunner
 \, Grigni\, Guibas\, Sharir\, and Welzl for general point sets in dimensio
 n $d\\geq 3$.\n
LOCATION:https://researchseminars.org/talk/CJCS/23/
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