A short proof of the first selecttion lemma and weak 1r -nets for moving points

Alexandre Rok, Shakhar Smorodinsky

Research output: Working paper/PreprintPreprint

Abstract

(i) We provide a short and simple proof of the first selection lemma. (ii) We also prove a selection lemma of a new type in Rd . For example, when d = 2 assuming n is large enough we prove that for any set
P of n points in general position there are Ω(n4) pairs of segments spanned by P all of which intersect in some fixed triangle spanned by P. (iii) Finally, we extend the weak 1r -net theorem to a kinetic setting where the underlying set of points is moving polynomially with bounded description complexity. We establish that one can find a kinetic analog N of a weak 1r -net of cardinality O (r d (d+1) 2 logd r) whose points are moving with coordinates that are rational functions with bounded description complexity. Moreover, each member of N has one polynomial coordinate.
Original languageEnglish GB
PublisherarXiv:1512.07505 [cs.DM]
StatePublished - 23 Dec 2015

Keywords

  • Computer Science - Discrete Mathematics
  • F.2.2
  • G.2.1
  • G.2.2

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