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We estimate the selection constant in the following geometric selectiontheorem by Pach: For every positive integer $d$ there is a constant $c_d > 0$such that whenever $X_1,..., X_{d+1}$ are $n$-element subsets of$\mathbb{R}^d$, then we can find a point $\mathbf{p} \in \mathbb{R}^d$ andsubsets $Y_i \subseteq X_i$ for every $i \in [d+1]$, each of size at least $c_dn$, such that $\mathbf{p}$ belongs to all {\em rainbow} $d$-simplicesdetermined by $Y_1,..., Y_{d+1}$, that is, simplices with one vertex in each$Y_i$.  We show a super-exponentially decreasing upper bound $c_d\leqe^{-(1/2-o(1))(d \ln d)}$. The ideas used in the proof of the upper bound alsohelp us prove Pach's theorem with $c_d \geq 2^{-2^{d^2 + O(d)}}$, which is alower bound doubly exponentially decreasing in $d$ (up to some polynomial inthe exponent). For comparison, Pach's original approach yields a triplyexponentially decreasing lower bound. On the other hand, Fox, Pach, and Sukrecently obtained a hypergraph density result implying a proof of Pach'stheorem with $c_d \geq2^{-O(d^2\log d)}$.  In our construction for the upper bound, we use the fact that the minimumsolid angle of every $d$-simplex is super-exponentially small. This fact waspreviously unknown and might be of independent interest. For the lower bound,we improve the "separation" part of the argument by showing that in one of thekey steps only $d+1$ separations are necessary, compared to $2^d$ separationsin the original proof.  We also provide a measure version of Pach's theorem.

Bounds for Pach's selection theorem and for the minimum solid angle in a simplex