Many Antipodal Pairs Force Many Neighboring Pairs
Abstract
Let X={x1,…,xn}⊂ ℝ2 be a finite set of points of diameter at most $1$. It is natural to expect that if many pairs (xi,xj) lie at distance close to $1$ from each other, then some clustering phenomenon must occur, implying that a significant number of these pairs are also very close to each other. %For 0<ε<1, we call a pair (xi,xj) ε-antipodal if |xi−xj|≥ 1−ε and ε-neighboring if |xi−xj|≤ ε. We prove that there exists a universal constant c>0 such that for all 0<ε<1, whenever n is large enough, we have: |{(i,j):|xi−xj|≤ ε}| ≥ c· ε1/2· |{(i,j):|xi−xj|≥ 1−ε}|. This confirms a recent conjecture of Steinerberger, who asked whether the ε1/2 ratio is the best possible. We also study a two-parameter version of Steinerberger's question by considering the number of pairs at distance at most ε1 and at distance at least 1−ε2. We show that in this case the optimal ratio is ε12·ε2−3/2 when ε1≤ε2, and ε11/2 otherwise. The proof proceeds by introducing an auxiliary graph associated with the set X and reducing the problem to bounding the largest eigenvalue of its adjacency matrix. Our main result is the outcome of human--AI interactions using ChatGPT 5.4.
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BibTeX
@misc{damasdi2026many,
title = {{Many Antipodal Pairs Force Many Neighboring Pairs}},
author = {Gábor Damásdi and Laurentiu Ploscaru},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.02605},
eprint = {2608.02605},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02605},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Many Antipodal Pairs Force Many Neighboring Pairs
AU - Gábor Damásdi
AU - Laurentiu Ploscaru
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2608.02605
UR - https://arxiv.org/abs/2608.02605
AB - Let X={x_1,…,x_n}⊂ ℝ^2 be a finite set of points of diameter at most $1$. It is natural to expect that if many pairs (x_i,x_j) lie at distance close to $1$ from each other, then some clustering phenomenon must occur, implying that a significant number of these pairs are also very close to each other. %For 0<ε<1, we call a pair (x_i,x_j) ε-antipodal if |x_i−x_j|≥ 1−ε and ε-neighboring if |x_i−x_j|≤ ε. We prove that there exists a universal constant c>0 such that for all 0<ε<1, whenever n is large enough, we have: |{(i,j):|x_i−x_j|≤ ε}| ≥ c· ε^1/2· |{(i,j):|x_i−x_j|≥ 1−ε}|. This confirms a recent conjecture of Steinerberger, who asked whether the ε^1/2 ratio is the best possible. We also study a two-parameter version of Steinerberger's question by considering the number of pairs at distance at most ε_1 and at distance at least 1−ε_2. We show that in this case the optimal ratio is ε_1^2·ε_2^−3/2 when ε_1≤ε_2, and ε_1^1/2 otherwise. The proof proceeds by introducing an auxiliary graph associated with the set X and reducing the problem to bounding the largest eigenvalue of its adjacency matrix. Our main result is the outcome of human--AI interactions using ChatGPT 5.4.
ER -