PreprintCC BY 4.0arXiv
Sharp Antipode-Neighbor Bounds in Every Dimension
Abstract
Let P⊂ℝd be finite with diam(P)≤1. For 0<ε<1, let 𝒩ε(P) denote the number of ordered pairs at distance at most ε, including the diagonal, and let 𝒜ε(P) denote the number of ordered pairs at distance at least 1−ε. We prove the sharp bound 𝒩ε(P)≳d ε(d−1)/2𝒜ε(P) conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform L2 estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel--Young gaps.
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BibTeX
@misc{korsky2026sharp,
title = {{Sharp Antipode-Neighbor Bounds in Every Dimension}},
author = {Samuel Korsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14037},
eprint = {2610.14037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14037},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Antipode-Neighbor Bounds in Every Dimension
AU - Samuel Korsky
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.14037
UR - https://arxiv.org/abs/2610.14037
AB - Let P⊂ℝ^d be finite with diam(P)≤1. For 0<ε<1, let 𝒩_ε(P) denote the number of ordered pairs at distance at most ε, including the diagonal, and let 𝒜_ε(P) denote the number of ordered pairs at distance at least 1−ε. We prove the sharp bound 𝒩_ε(P)≳_d ε^(d−1)/2𝒜_ε(P) conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform L^2 estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel--Young gaps.
ER -