Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains
Abstract
Let Ω be a bounded convex domain in the plane and let R be a minimum-area rectangle containing Ω̅. We prove that if the ratio of the shorter side of R to its longer side is at most ρ≈ 0.443485, then the first nonzero Neumann eigenvalue of Ω is bounded from below by that of R. Moreover, equality holds if and only if Ω coincides with the interior of R. The proof is based on a flux identity for the means of a test function over vertical sections of Ω, combined with weighted estimates involving the distributional second derivatives of the boundary functions. Finally, we show that the threshold ρ≈0.443485 is nearly optimal by constructing, for every ratio in $(0.457994,1/2)$, trapezoids that are arbitrarily close to their minimum-area enclosing rectangle and for which the inequality fails.
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BibTeX
@misc{li2026minimumarea,
title = {{Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains}},
author = {Xiaolong Li and Jin Sun and Kui Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23439},
eprint = {2610.23439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23439},
note = {CC0},
}
RIS
TY - UNPB
TI - Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains
AU - Xiaolong Li
AU - Jin Sun
AU - Kui Wang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23439
UR - https://arxiv.org/abs/2610.23439
AB - Let Ω be a bounded convex domain in the plane and let R be a minimum-area rectangle containing Ω̅. We prove that if the ratio of the shorter side of R to its longer side is at most ρ≈ 0.443485, then the first nonzero Neumann eigenvalue of Ω is bounded from below by that of R. Moreover, equality holds if and only if Ω coincides with the interior of R. The proof is based on a flux identity for the means of a test function over vertical sections of Ω, combined with weighted estimates involving the distributional second derivatives of the boundary functions. Finally, we show that the threshold ρ≈0.443485 is nearly optimal by constructing, for every ratio in $(0.457994,1/2)$, trapezoids that are arbitrarily close to their minimum-area enclosing rectangle and for which the inequality fails.
ER -