  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
$|W_1|\cdot\ldots\cdots|W_k|=\Omega\left(\varepsilon^{d(k-1)+1}|V_1|\cdot
|V_2|\cdot\ldots\cdot|V_k|\right)$.
