[Paper Review] On convex holes in $d$-dimensional point sets

Boris Bukh, Ting-Wei Chao+1 more|arXiv (Cornell University)|Jul 17, 2020

Mathematical Approximation and Integration15 references4 citations

TL;DR

This paper improves the upper bound on the size of the largest hole in d-dimensional point sets in general position, showing that arbitrarily large sets exist without holes of size O(4^d d log d) or larger. Using a novel construction based on relaxed (t,m,s)-nets in base 2, the authors achieve a bound of 2^{7d}, significantly improving upon Valtr's previous d^{d+o(d)} bound and providing tighter estimates for h(d).

ABSTRACT

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of $(t,m,s)$-nets, obeying a relaxed equidistribution condition.

Motivation & Objective

• To improve the known upper bound on the maximum size of a hole in d-dimensional point sets in general position.
• To address the longstanding open problem of determining h(d), the largest ℓ for which every large enough d-dimensional set in general position contains an ℓ-hole.
• To construct point sets in R^d with no large holes, using combinatorial and discrepancy-theoretic tools.
• To refine Valtr's earlier bound of d^{d+o(d)} by replacing the prime product structure with a more efficient equidistribution framework based on base-2 nets.

Proposed method

• Constructing point sets using (t,m,s)-nets in base 2, which are known from discrepancy theory for uniform distribution.
• Introducing a relaxed equidistribution condition that allows smaller quality parameters t, reducing the hole size bound.
• Applying a generalized version of Horton's construction in higher dimensions, using the structure of (t,s)-sequences in base 2.
• Using a projection and convex hull argument to show that no large convex polytope can be empty in the constructed sets.
• Leveraging the Chinese remainder theorem and dyadic sub-boxes to ensure uniform distribution of points across the space.
• Proving that if a point lies in the relative interior of a convex hull of subsets, then it cannot be part of a large empty convex polytope.

Experimental results

Research questions

• RQ1 What is the largest possible size of a hole in a d-dimensional point set in general position?
• RQ2 Can the exponential bound on h(d) be improved beyond the d^{d+o(d)} result of Valtr?
• RQ3 Can constructions based on (t,m,s)-nets in base 2 yield better bounds on hole-free sets than previous prime-based methods?
• RQ4 Is it possible to construct arbitrarily large d-dimensional point sets without holes of size O(4^d d log d) or larger?
• RQ5 Do constructions based on base-2 nets avoid the superexponential dependence on prime products seen in Valtr’s method?

Key findings

• The paper establishes a new upper bound: h(d) < 2^{7d}, improving upon Valtr’s d^{d+o(d)} bound.
• For specific dimensions, the bound is significantly tighter: h(3) ≤ 32, h(4) ≤ 240, h(5) ≤ 988, h(6) ≤ 8000.
• The improved bound is achieved using a variant of (t,m,s)-nets with a relaxed equidistribution condition, avoiding the need for large prime factors.
• The construction uses base-2 (t,s)-sequences and their truncations to generate point sets with controlled distribution and no large empty convex polytopes.
• The authors show that any set constructed via this method cannot contain a hole larger than O(4^d d log d), which is asymptotically better than previous results.
• The method demonstrates that the superexponential term in Valtr’s bound arises from the use of multiple prime moduli, which can be replaced by a single base-2 structure with better scalability.