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A Sharp Lower Bound for Integral Menger Curvature
Abstract
We prove that round circles CL are the unique minimisers of integral Menger curvature among embedded closed C1 curves γ of prescribed length L for every p≥ 2, i.e. Mp(CL) ≤ Mp(γ), with equality if and only if γ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For p>3, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals L. In particular, Mp detects the unknot and is basic for p>3, thereby answering a question posed repeatedly in the literature.
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BibTeX
@misc{dohmen2026sharp,
title = {{A Sharp Lower Bound for Integral Menger Curvature}},
author = {Alexander Dohmen and Sebastian Scholtes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15981},
eprint = {2610.15981},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15981},
}
RIS
TY - UNPB
TI - A Sharp Lower Bound for Integral Menger Curvature
AU - Alexander Dohmen
AU - Sebastian Scholtes
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.15981
UR - https://arxiv.org/abs/2610.15981
AB - We prove that round circles C_L are the unique minimisers of integral Menger curvature among embedded closed C^1 curves γ of prescribed length L for every p≥ 2, i.e. M_p(C_L) ≤ M_p(γ), with equality if and only if γ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For p>3, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals L. In particular, M_p detects the unknot and is basic for p>3, thereby answering a question posed repeatedly in the literature.
ER -