Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices in $P$, for $0<\epsilon\leq 1$. We show that there is a point $x\in {\mathbb R}^d$ that pierces $\displaystyle \Omega\left(\epsilon^{(d^4+d)(d+1)+\delta}{n\choose d+1}\right)$ simplices in $E$, for any fixed $\delta>0$. This is a dramatic improvement in all dimensions $d\geq 3$, over the previous lower bounds of the general form $\displaystyle \epsilon^{(cd)^{d+1}}n^{d+1}$, which date back to the seminal 1991 work of Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman. As a result, any $n$-point set in general position in $\mathbb{R}^d$ admits only $\displaystyle O\left(n^{d-\frac{1}{d(d-1)^4+d(d-1)}+\delta}\right)$ halving hyperplanes, for any $\delta>0$, which is a significant improvement over the previously best known bound $\displaystyle O\left(n^{d-\frac{1}{(2d)^{d}}}\right)$ in all dimensions $d\geq 5$. An essential ingredient of our proof is the following semi-algebraic Tur\'an-type result of independent interest: Let $(V_1,\ldots,V_k,E)$ be a hypergraph of bounded semi-algebraic description complexity in ${\mathbb R}^d$ that satisfies $|E|\geq \varepsilon |V_1|\cdot\ldots \cdot |V_k|$ for some $\varepsilon>0$. Then there exist subsets $W_i\subseteq V_i$ that satisfy $W_1\times W_2\times\ldots\times W_k\subseteq E$, and $|W_1|\cdot\ldots\cdots|W_k|=\Omega\left(\varepsilon^{d(k-1)+1}|V_1|\cdot |V_2|\cdot\ldots\cdot|V_k|\right)$.

翻译：设 $(P,E)$ 是一个 $(d+1)$-均匀几何超图，其中 $P$ 是 $\mathbb{R}^d$ 中处于一般位置的 $n$ 个点构成的集合，$E\subseteq {P\choose d+1}$ 是顶点在 $P$ 中的 $\epsilon{n\choose d+1}$ 个 $d$ 维单形构成的集合，这里 $0<\epsilon\leq 1$。我们证明存在一点 $x\in {\mathbb R}^d$ 穿透了 $E$ 中 $\displaystyle \Omega\left(\epsilon^{(d^4+d)(d+1)+\delta}{n\choose d+1}\right)$ 个单形，其中 $\delta>0$ 为任意给定常数。这一结果在所有维度 $d\geq 3$ 上显著改进了先前可追溯到 Alon、Bárány、Füredi 和 Kleitman 1991 年开创性工作的形如 $\displaystyle \epsilon^{(cd)^{d+1}}n^{d+1}$ 的一般性下界。由此可得，在 $\mathbb{R}^d$ 中处于一般位置的任意 $n$ 个点仅包含 $\displaystyle O\left(n^{d-\frac{1}{d(d-1)^4+d(d-1)}+\delta}\right)$ 个平分超平面（$\delta>0$ 为任意常数），这显著改进了在所有维度 $d\geq 5$ 上先前最优的 $\displaystyle O\left(n^{d-\frac{1}{(2d)^{d}}}\right)$ 估计。我们证明的一个关键要素是如下具有独立意义的半代数 Turán 型结果：设 $(V_1,\ldots,V_k,E)$ 是 ${\mathbb R}^d$ 中具有有界半代数描述复杂度的超图，且对某 $\varepsilon>0$ 满足 $|E|\geq \varepsilon |V_1|\cdot\ldots \cdot |V_k|$。则存在子集 $W_i\subseteq V_i$ 使得 $W_1\times W_2\times\ldots\times W_k\subseteq E$，且 $|W_1|\cdot\ldots\cdots|W_k|=\Omega\left(\varepsilon^{d(k-1)+1}|V_1|\cdot |V_2|\cdot\ldots\cdot|V_k|\right)$。