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.

翻译：对于 $\mathbb R^d$（$d\ge 2$）中一个包含 $n$ 个点的集合 $P$，若超平面 $h$ 包含 $P$ 中至少 $k$ 个点，则称该超平面关于 $P$ 是 $k$-富有的。为回答并推广 Peyman Afshani 提出的问题，我们证明：若 $\mathbb R^d$（$d \geq 3$）中 $k$-富有超平面的数量至少为 $\Omega(n^d/k^\alpha + n/k)$（比例常数足够大，且满足 $d\le \alpha < 2d-1$），则存在一个 $(d-2)$-平坦包含 $P$ 中 $\Omega(k^{(2d-1-\alpha)/(d-1)})$ 个点。我们还给出了上界构造实例，表明上述下界是紧的。通过扩展分析，我们进一步得到了关于 $k$-富有球面或 $k$-富有平坦的类似下界。