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(i) We provide a short and simple proof of the first selection lemma. (ii) Wealso 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 ingeneral position there are $\Omega(n^4)$ pairs of segments spanned by $P$ allof which intersect in some fixed triangle spanned by $P$. (iii) Finally, weextend the weak $\frac{1}{r}$-net theorem to a kinetic setting where theunderlying set of points is moving polynomially with bounded descriptioncomplexity. We establish that one can find a kinetic analog $N$ of a weak$\frac{1}{r}$-net of cardinality $O(r^{\frac{d(d+1)}{2}}\log^{d}r)$ whosepoints are moving with coordinates that are rational functions with boundeddescription complexity. Moreover, each member of $N$ has one polynomialcoordinate.

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