A short proof of the first selection lemma and weak $\frac{1}{r}$-nets for moving points

TL;DR

This paper presents a concise proof of the first selection lemma, introduces a new selection lemma for moving points in ^d, and extends the weak r-net theorem to a kinetic setting with moving points described by rational functions.

Contribution

It provides a simplified proof of the first selection lemma, introduces a novel selection lemma for moving points, and extends weak r-nets to polynomially moving points in a kinetic setting.

Findings

A short proof of the first selection lemma.

A new selection lemma involving intersecting segments in ^d.

A kinetic r-net with size bounds for moving points.

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 $\Re^d$. 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 $\Omega(n^4)$ pairs of segments spanned by $P$ all of which intersect in some fixed triangle spanned by $P$. (iii) Finally, we extend the weak $\frac{1}{r}$-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 $\frac{1}{r}$-net of cardinality $O(r^{\frac{d(d+1)}{2}}\log^{d}r)$ whose points are moving with coordinates that are rational functions with bounded description complexity. Moreover, each member of $N$ has one polynomial coordinate.