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Many antipodes implies many neighbors

Stefan Steinerberger

TL;DR

This work addresses how many near-antipodal pairs force many near-neighbor pairs in planar point sets of diameter at most 1. It proves a universal lower bound: the number of ε-neighbors is at least a constant times $\varepsilon^{3/4}/(\log \varepsilon^{-1})^{1/4}$ times the number of antipodal pairs, with the optimal rate conjectured to be $\varepsilon^{1/2}$ and demonstrated by explicit examples. The authors develop a proof that combines geometric reductions to an ε-neighborhood of the boundary, a box-tiling discretization, and a linear-algebra bound via a matrix of antipodal possibilities, together with a graph-theoretic lemma to bound edge counts; the bound holds up to a logarithmic factor. They also discuss higher-dimensional extensions, provide sharp constructions achieving the conjectured rate, and place the result in the context of classical diameter-distance problems in extremal geometry. The methods offer a framework linking combinatorial geometry, spectral bounds, and graph-theoretic techniques to quantify local-global relationships in geometric configurations.

Abstract

Suppose $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\leq 1$, meaning $\|x_i - x_j\| \leq 1$ for all $1 \leq i,j \leq n$. We show that if there are many `antipodes', these are pairs of points of with distance $\geq 1-\varepsilon$, then there are many neighbors, these are pairs of points that are distance $\leq \varepsilon$. More precisely, we prove that for some universal $c>0$, $$ \# \left\{(i,j): \|x_i - x_j\| \leq \varepsilon\right\} \geq \frac{c \cdot \varepsilon^{3/4}}{\left( \log \varepsilon^{-1} \right)^{1/4}}\cdot \# \left\{(i,j): \|x_i - x_j\| \geq 1- \varepsilon\right\}.$$ The inequality is very easy too prove with factor $\varepsilon^2$ and easy with $\varepsilon$. The optimal rate might be $\varepsilon^{1/2}$ which is attained by several examples.

Many antipodes implies many neighbors

TL;DR

This work addresses how many near-antipodal pairs force many near-neighbor pairs in planar point sets of diameter at most 1. It proves a universal lower bound: the number of ε-neighbors is at least a constant times times the number of antipodal pairs, with the optimal rate conjectured to be and demonstrated by explicit examples. The authors develop a proof that combines geometric reductions to an ε-neighborhood of the boundary, a box-tiling discretization, and a linear-algebra bound via a matrix of antipodal possibilities, together with a graph-theoretic lemma to bound edge counts; the bound holds up to a logarithmic factor. They also discuss higher-dimensional extensions, provide sharp constructions achieving the conjectured rate, and place the result in the context of classical diameter-distance problems in extremal geometry. The methods offer a framework linking combinatorial geometry, spectral bounds, and graph-theoretic techniques to quantify local-global relationships in geometric configurations.

Abstract

Suppose is a set of points in the plane with diameter , meaning for all . We show that if there are many `antipodes', these are pairs of points of with distance , then there are many neighbors, these are pairs of points that are distance . More precisely, we prove that for some universal , The inequality is very easy too prove with factor and easy with . The optimal rate might be which is attained by several examples.

Paper Structure

This paper contains 14 sections, 4 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction and Statement
  2. Introduction
  3. Result.
  4. Examples.
  5. Higher dimensions and metric spaces.
  6. Related results
  7. Proof
  8. Outline of the argument.
  9. Reducing to $\varepsilon$ small.
  10. Reducing to a neighborhood of the convex hull.
  11. Linear Algebra.
  12. Linear Algebra
  13. Some Graph Theory
  14. Conclusion of the Argument

Key Result

Theorem 1

There exists a universal constant $c>0$ such that for all $\varepsilon >0$ and any set $\left\{x_1, \dots, x_n \right\} \subset \mathbb{R}^2$ with diameter $\leq 1$, i.e. satisfying $\|x_i -x_j\| \leq 1$,

Figures (5)

  • Figure 1: $n$ points with $\sim n^2$ pairs of points at distance $\geq 1 - \varepsilon$ but also $\sim n^2$ pairs of points at distance $\leq \varepsilon$.
  • Figure 2: Left: a Reuleaux triangle with many points in the vertices, right: a regular polygon with points evenly spaced.
  • Figure 3: A sketch of a type of metric space where no such property holds: there can be many antipodes, pairs of points at distance $\geq 2$ while any pair of points is at least $1-$separated.
  • Figure 4: Reducing to a $\varepsilon-$neighborhood of the boundary and then breaking it into $\varepsilon/4 \times \varepsilon/4$ boxes.
  • Figure 5: A cartoon picture of the argument.

Theorems & Definitions (7)

  • Theorem
  • Lemma
  • proof
  • Lemma
  • proof
  • Lemma
  • proof