Source-linked AI summary
Helly-Type Theorems for Splitting Point Sets
Lidor Portal, Natan Rubin
TL;DR
The paper asks how Helly-type conditions can control splitting of many finite point sets, especially beyond point and hyperplane transversals. It develops direct and approximate (p, q)-type results for k-flats, obtaining splitting guarantees for every dimension 0≤k≤d−1 while identifying limits on the approach and scope.
Problem
Existing Helly-type transversal criteria do not address families of many point sets split by k-flats of arbitrary dimensionality, and intermediate-dimensional convex transversals lack comparable (p,q)-theorems.
Method
The paper formulates a (p,q,α)-property for splitting point sets and uses convex-hull, centerset, ε-approximation, and bounded-VC-dimension constructions to obtain k-flat transversals.
Results
The results provide finite families of splitting k-flats for every 0≤k≤d−1; for hyperplanes, at most C hyperplanes achieve β-splitting, while generally at most C k-flats achieve (α−ε)-splitting.
Takeaways & Limitations
Helly-type conditions can control splitting families of finite point sets beyond the classical point and hyperplane transversal settings, with an approximate guarantee for arbitrary k-flat dimensions.
Takeaways & Limitations
The general k-flat theorem requires ε>0, and the paper leaves open whether the hyperplane result can achieve β=α; its proof does not extend directly to intermediate-dimensional k-flats.
Abstract
from arXiv · showhide
Let $0 < α\leq 1/2$. We say that a finite point set $P$ in $\mathbb{R}^d$ is $α$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $α|P|$ of the points of $P$. We further say $P$ is $α$-split by a $k$-dimensional flat $τ$ if $P$ is $α$-split by any hyperplane through $τ$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $τ$ has depth $α$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.
1 Introduction
The paper extends Helly-type questions from convex-set transversals to splitting families of finite point sets, including k-flats of arbitrary dimension. It addresses settings with many point sets where a single hyperplane may split only a few members.
- A k-flat is a translate of a k-dimensional linear subspace; points and hyperplanes are the cases k=0 and k=d−1.
- For n≫d point sets in small balls with centers in general position, no hyperplane can split more than d sets simultaneously.The obstruction follows because a splitting hyperplane must cross the corresponding convex hulls.
- The intermediate-dimensional k-flat case cannot use the classical convex-transversal (p,q) theorems, because comparable theorems fail for general convex-set families when 1≤k≤d−2.
- For hyperplanes, families with this property can be β-split by at most C(p, q, d) hyperplanes, where β=β(α,d)∈(0,α).
- More generally, for 0≤k≤d−1 and q≥(k+1)(d−k)+1, the family can be (α−ϵ)-split by at most C(p,q,d,k,ϵ) k-flats.The guarantee applies for α∈(0,1/2] and ϵ∈(0,α).
2 Preliminaries
The preliminaries introduce centerpoints, Helly and fractional Helly numbers, geometric range spaces, VC-dimension tools, and semialgebraic-set bounds used to derive transversal results.
- Centerpoints: A centerpoint is a point such that every closed half-space containing it contains at least |X|/(d + 1) points of X.
- Centerpoints: Rado’s Center Point Theorem guarantees that every finite point set in R^d has a centerpoint.
- Helly and fractional Helly numbers: The Helly number is the smallest k for which intersecting every k-member subfamily guarantees a common intersection, while the fractional Helly number uses a positive fraction of intersecting k-tuples.
- Geometric range spaces: A range space consists of geometric objects X and a family F of subsets of X; its VC-dimension is the largest shattered-subset size, and its shatter function counts realized restrictions.
- Geometric range spaces: The dual shatter function counts nonempty Venn-diagram fields, and polynomial primal and dual shatter functions are equivalent.
- Range-space tools: Bounded VC-dimension supplies small ε-nets and ε-approximations, while bounded dual shatter growth yields a (p, q)-theorem with a constant-size transversal.
3 Proof of Theorem 1.3
The proof constructs a compact convex (α, β)-core for each point set, then reduces splitting point sets to transversals of convex cores and applies an almost-Helly theorem for hyperplanes.
- Core construction: For 0 < α ≤ 1/2, every point set P in R^d has a compact convex (α, β)-core for some β = β(α, d) in (0, α).An (α, β)-core is intersected by every k-flat that α-splits P, while every k-flat intersecting it β-splits P.
- Core construction: The construction partitions P into at most r parts enclosed by simplices, with each part between |P|/r and 2|P|/r and at most b r^(1−1/d) simplices crossed by any hyperplane.This uses Matoušek’s simplicial partition theorem.
- Core construction: A transversal T selects one point from each partition part, and the core is the convex hull of centerpoints assigned to all s-subsets of T, where s = ⌈αr/2⌉.Rado’s centerpoint theorem ensures the constructed core is nonempty; finiteness makes it compact and convex.
- Core properties: If a hyperplane through an α-splitting flat avoided the core, centerpoint depth would force it to cross too many enclosing simplices, contradicting the simplicial partition bound.The contradiction arises because at least s/(d + 1) simplices must be crossed, while the partition permits at most b r^(1−1/d).
- Theorem 1.3: Applying the hyperplane (p, q)-theorem to the family of cores yields C(p, q, d) hyperplanes that intersect every core and therefore β-split every corresponding point set.Any q point sets that are α-split by one hyperplane give q cores crossed by that hyperplane, allowing Theorem 1.2 to be applied.
4 Proof of Theorem 1.4
The proof encodes approximately splitting hyperplanes and k-flats as bounded-complexity semi-algebraic sets, then applies a semi-algebraic transversal theorem in parameter space. This yields a bounded collection of k-flats that approximately splits the required point-set families.
- Hyperplane encoding: A semi-algebraic set Γα,ε(P) represents hyperplanes: every α-splitting hyperplane maps into Γα,ε(P), while every represented point defines an (α−ε)-splitting hyperplane.Its description complexity depends only on d and ε.
- Hyperplane encoding: An ε-approximation A preserves α-splitting up to ε: an α-splitting k-flat for P becomes (α−ε)-splitting for A, and conversely.The approximation is obtained from the range space of open halfspaces, whose VC-dimension is d+1.
- From hyperplanes to k-flats: A generic k-flat is parameterized by an ordered tuple of k+1 points chosen from fixed complementary (d−k)-flats, giving a parameter space of dimension (k+1)(d−k).For k=1 in R^3, the representation uses 4 parameters.
- Applying the transversal theorem: For q=(k+1)(d−k)+1, the Λ-family has the (p,q)-property, so a semi-algebraic transversal theorem supplies at most C parameter points.The constant depends only on d, k, p, and ε, and each parameter point corresponds to a k-flat.
- From hyperplanes to k-flats: Λα,ε,k(Pi) consists of tuples whose affine-span k-flat avoids the origin and whose hyperplanes through it lie in Γα,ε(Pi).This transfers the splitting condition from hyperplanes to k-flats in the parameter space.
- Applying the transversal theorem: The resulting bounded transversal in parameter space corresponds to a fixed number of k-flats that (α−ε)-split the relevant point sets.The bound depends only on the ambient and theorem parameters, not on the family size.
5 Concluding Remarks
The paper gives two proofs of Theorem 1.3, but only one extends to intermediate-dimensional flats. The authors note that exact generalizations and further quantitative Helly-type questions remain open.
- Only one of the two proofs of Theorem 1.3 extends to flats with 1≤k≤d−2.
- A general (p,q)-statement for k-flat transversals is unlikely to hold with ε=0, because such theorems fail for general families of convex sets.
- The authors leave open whether Theorem 1.3 can be proved with β=α and whether their ideas apply to other quantitative or approximate Helly-type problems.