Publication

Tight lower bounds for the size of epsilon-nets

János Pach, Gábor Tardos
2011
Conference Papers

Abstract

According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an epsilon-net of size O (1/epsilon log 1/epsilon). 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 epsilon-nets is superlinear in 1/epsilon, were found by Alon (2010). In his examples, every epsilon-net is of size Omega (1/epsilon g(1/epsilon)), where g is an extremely slowly growing function, related to the inverse Ackermann function.

Official source

https://infoscience.epfl.ch/entities/publication/3e484089-4241-4c15-9436-d2176cca6ab8

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János Pach, Gábor Tardos
2011
Conference Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/3e484089-4241-4c15-9436-d2176cca6ab8

About this result

Related concepts (20)

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced ˈbanax) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.

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