Tight lower bounds for the size of epsilon-nets

Pach, János; Tardos, Gábor

doi:10.1090/S0894-0347-2012-00759-0

Tight lower bounds for the size of epsilon-nets
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by János Pach and Gábor Tardos

J. Amer. Math. Soc. 26 (2013), 645-658

DOI: https://doi.org/10.1090/S0894-0347-2012-00759-0

Published electronically: November 7, 2012

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Abstract:

According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an $\varepsilon$-net of size $O\!\left (\frac {1}{\varepsilon }\log \frac 1{\varepsilon }\right )$. Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest $\varepsilon$-nets is superlinear in $\frac 1{\varepsilon }$, were found by Alon (2010). In his examples, every $\varepsilon$-net is of size $\Omega \left (\frac {1}{\varepsilon }g(\frac {1}{\varepsilon })\right )$, where $g$ is an extremely slowly growing function, related to the inverse Ackermann function.

We show that there exist geometrically defined range spaces, already of VC-dimension $2$, in which the size of the smallest $\varepsilon$-nets is $\Omega \left (\frac {1}{\varepsilon }\log \frac {1}{\varepsilon }\right )$. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest $\varepsilon$-nets is $\Omega \left (\frac {1}{\varepsilon }\log \log \frac {1}{\varepsilon }\right )$. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

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