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 $\mathbb R^d$, $d \geq 3$, is at least $\Omega(n^d/k^\alpha + n/k)$, with a sufficiently large constant of proportionality and with $d\le \alpha < 2d-1$, then there exists a $(d-2)$-flat that contains $\Omega(k^{(2d-1-\alpha)/(d-1)})$ 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.