One-Sided Epsilon-Approximants



Given a finite point set \(P \subset \mathbb{R}^{d}\), we call a multiset A a one-sided weak ɛ-approximant for P (with respect to convex sets), if \(\vert P \cap C\vert /\vert P\vert -\vert A \cap C\vert /\vert A\vert \leq \varepsilon\) for every convex set C.

We show that, in contrast with the usual (two-sided) weak ɛ-approximants, for every set \(P \subset \mathbb{R}^{d}\) there exists a one-sided weak ɛ-approximant of size bounded by a function of ɛ and d.



We thank the referees for pointing omissions and typos. We are also grateful to Po-Shen Loh and the late Jirka Matoušek for discussions.


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Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer ScienceAriel UniversityArielIsrael

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