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Weinstock Inequality on Regular Trees
Abstract
Let Tn be the infinite n-regular tree, n≥3. We prove that every finite connected vertex set Ω⊂ Tn satisfies the sharp inequality σ1(Ω)≤ n/((n−1)|Ω|+1). Equality holds if and only if Ω is a ball. Since |δΩ|=(n−2)|Ω|+2, the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on Tn.
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BibTeX
@misc{wang2026weinstock,
title = {{Weinstock Inequality on Regular Trees}},
author = {Lili Wang and Tao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06067},
eprint = {2609.06067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06067},
}
RIS
TY - UNPB
TI - Weinstock Inequality on Regular Trees
AU - Lili Wang
AU - Tao Wang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06067
UR - https://arxiv.org/abs/2609.06067
AB - Let T_n be the infinite n-regular tree, n≥3. We prove that every finite connected vertex set Ω⊂ T_n satisfies the sharp inequality σ_1(Ω)≤ n/((n−1)|Ω|+1). Equality holds if and only if Ω is a ball. Since |δΩ|=(n−2)|Ω|+2, the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on T_n.
ER -