TY - GEN

T1 - Stronger bounds for weak epsilon-nets in higher dimensions

AU - Rubin, Natan

N1 - Funding Information:
and p larger than a certain constant which depends on γ . 4. Our proof of Theorem 1.1 is fully constructive; for details see the full paper [44]. ACKNOWLEDGMENTS The project leading to this application has received funding from European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 678765. Also supported by Ralph Selig Career Development Chair in Information Theory.
Publisher Copyright:
© 2021 ACM.

PY - 2021/6/15

Y1 - 2021/6/15

N2 - Given a finite point set P in Rd, and >0 we say that N? Rd is a weak -net if it pierces every convex set K with |K P|? ? |P|. Let d? 3. We show that for any finite point set in Rd, and any ?>0, there exist a weak -net of cardinality O(1/?d-1/2+?), where ?>0 is an arbitrary small constant. This is the first improvement of the bound of O?(1/?d) that was obtained in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension d? 3.

AB - Given a finite point set P in Rd, and >0 we say that N? Rd is a weak -net if it pierces every convex set K with |K P|? ? |P|. Let d? 3. We show that for any finite point set in Rd, and any ?>0, there exist a weak -net of cardinality O(1/?d-1/2+?), where ?>0 is an arbitrary small constant. This is the first improvement of the bound of O?(1/?d) that was obtained in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension d? 3.

KW - computational geometry

KW - convex sets

KW - epsilon-nets

KW - selection theorems

UR - http://www.scopus.com/inward/record.url?scp=85108140212&partnerID=8YFLogxK

U2 - 10.1145/3406325.3451062

DO - 10.1145/3406325.3451062

M3 - Conference contribution

AN - SCOPUS:85108140212

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 989

EP - 1002

BT - STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

A2 - Khuller, Samir

A2 - Williams, Virginia Vassilevska

PB - Association for Computing Machinery

T2 - 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021

Y2 - 21 June 2021 through 25 June 2021

ER -