Covering points by hyperplanes and related problems

For a set $P$ of $n$ points in ${\mathbb{R}}^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of $k$-rich hyperplanes in ${\mathbb{R}}^d$, $d\ge 3$, is at least $\mathrm{\Omega }(n^d\mathrm{/}{k}^{\alpha }+n\mathrm{/}k)$, with a sufficiently large constant of proportionality and with $d\le \alpha <2d-1$, then there exists a $(d-2)$-flat that contains $\mathrm{\Omega }(k^{(2d-1-\alpha )\mathrm{/}(d-1)})$ points of $P$. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for $k$-rich spheres or $k$-rich flats.