Improved Bounds for Point Selections and Halving Hyperplanes in Higher Dimensions

TL;DR

This work advances the study of point selections in higher-dimensional geometric hypergraphs by introducing a top-down partition strategy combined with a sharp semi-algebraic Turán-type theorem. The authors obtain a dramatically improved exponent for the Second Selection Theorem in all dimensions $d\ge3$, showing that a piercing point hits at least $\Omega\left({\varepsilon}^{(d^4+d)(d+1)+\delta}{n\choose d+1}\right)$ edges, for any fixed $\delta>0$. They derive a substantially stronger bound on the number of halving hyperplanes (and more generally $k$-sets) in dimension $d\ge5$, via a tight connection between selection and halving quantities. A core technical contribution is an efficient semi-algebraic Turán-type result that guarantees large complete subproducts in dense semi-algebraic hypergraphs, enabling the top-down charging scheme that drives the main bounds. The results unify probabilistic-geometric partition methods with algebraic-geometry tools to yield improved upper bounds and offer new structural insights for semi-algebraic hypergraphs and their applications to discrete geometry problems like halving and $k$-sets.

Abstract

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 $ε{n\choose d+1}$ $d$-dimensional simplices with vertices in $P$, for $0<ε\leq 1$. We show that there is a point $x\in {\mathbb R}^d$ that pierces $\displaystyle Ω\left(ε^{(d^4+d)(d+1)+δ}{n\choose d+1}\right)$ simplices in $E$, for any fixed $δ>0$. This is a dramatic improvement in all dimensions $d\geq 3$, over the previous lower bounds of the general form $\displaystyle ε^{(cd)^{d+1}}n^{d+1}$, which date back to the seminal 1991 work of Alon, Bárány, Füredi and Kleitman. As a result, any $n$-point set in general position in $\mathbb{R}^d$ admits only $\displaystyle O\left(n^{d-\frac{1}{d(d-1)^4+d(d-1)}+δ}\right)$ halving hyperplanes, for any $δ>0$, 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 Turán-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 $|W_1|\cdot\ldots\cdots|W_k|=Ω\left(\varepsilon^{d(k-1)+1}|V_1|\cdot |V_2|\cdot\ldots\cdot|V_k|\right)$.

Figures (5)

• Figure 1: A tight triple of simplices $\{\Delta_1, \Delta_2,\Delta_3\}\subset \Sigma$ in ${\mathbb R}^2$, with enclosed subsets $P_1,P_2,P_3$. The point $x\in {\mathbb R}^3$ pierces every triangle $\triangle p_1p_2p_3$, with vertices $p_1\in P_1,p_2\in P_2$ and $p_3\in P_3$.
• Figure 2: The convex sets $K_1, K_2$ and $K_3$ comprise a tight triple ${{\mathcal{K}}}$ in ${\mathbb R}^2$. Left: The convex hulls $C_i=C\left({{\mathcal{K}}}\setminus \{K_i\}\right)$ are depicted for $1\leq i\leq 3$. Note that we have $\bigcap_{i=1}^3C_i=\emptyset$. Right: The simplex $\Delta({{\mathcal{K}}})=\bigcap H_i^-$ is depicted along with the inner tangents $H_i=H\left({{\mathcal{K}}}\setminus \{K_i\},\{K_i\}\right)$. By Theorem \ref{['Theorem:StronglySeparated']}, if ${{\mathcal{K}}}$ is tight, then $\Delta({{\mathcal{K}}})$ is non-empty, and is contained in every triangle $\triangle x_1x_2x_3$ with $x_i\in K_i$, for $1\leq i\leq 3$.
• Figure 3: Left: The family ${{\mathcal{K}}}=\{K_1,K_2,K_3\}$ in ${\mathbb R}^2$ yields $\Delta({{\mathcal{K}}})=\bigcap_{i=1}^{d+1} H_i^-=\emptyset$ so that, by Theorem \ref{['Theorem:StronglySeparated']}, ${{\mathcal{K}}}$ is loose. Right: The point sequence $\left(x_1,x_2,x_{3}\right)$ in ${\mathbb R}^2$ is split by the sequence $\left(G_1,G_2,G_{3}\right)$ of hyperplanes. The $2$-simplex (i.e., triangle) $\Delta=\bigcap_{i=1}^{3}G_i^-$ is depicted.
• Figure 4: Lemma \ref{['Lemma:PinHyperplane']} in dimension $d=2$: moving the hyperplane $H$ crossing the simplices (i.e., triangles) $\Delta_1,\Delta_2$ and $\Delta_3$ to a position in which it contains a pair of vertices of $V(\{\Delta_1,\Delta_2,\Delta_3\})$ while still intersecting each $\Delta_i$.
• Figure 5: The loose family ${{\mathcal{K}}}=\{\Delta_1,\ldots,\Delta_{d+1}\}$ of simplices in general position within ${\mathbb R}^d$ is pinned by a point $x\in C({{\mathcal{K}}})$. By Lemma \ref{['Claim:HyperplaneThroughPoint']}, there is a hyperplane $H$ through $x$ that crosses at least $d$ elements of ${{\mathcal{K}}}$.

Theorems & Definitions (51)

• Theorem 1.1: BorosFurediBarany
• Theorem 1.2: Alon-Bárány-Füredi-Kleitman AlonSelections, 1990
• Theorem 1.3: BFL
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1: Pach's theorem PachTheorem
• Theorem 2.2: The Colored Selection Theorem of Karasev Karasev
• Lemma 2.3
• proof