A short open space curve of unit width
Abstract
In this work, we propose a curve to be a minimum of length among rectifiable, continuous curves of unit width in three-dimensional Euclidean space. We show that a previously conjectured minimizer of that problem is not of unit width, give the numerical derivation of a candidate family of unit width curves, prove that it admits a strict local minimum within that candidate family and that this minimum is of unit width and length 3.905124162243593…. The curve consists of eight straight segments, two cone geodesics, and three circular helices. Its parameters are specified by a uniquely isolated solution of the joining and stationarity equations. To establish unit width in every direction, we combine analytic estimates with interval arithmetic on a finite covering of the remaining directions. We conjecture that the curve is globally minimizing. We supplement this work with the programs used to identify the candidates, replicate the computer-assisted parts of our proof, as well as a formalization of our proof in Lean.
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BibTeX
@misc{bartels2026short,
title = {{A short open space curve of unit width}},
author = {Tinko Sebastian Bartels and Dennis Kevin Choy and Ruixiang Wu and Tongxin Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19585},
eprint = {2610.19585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19585},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A short open space curve of unit width
AU - Tinko Sebastian Bartels
AU - Dennis Kevin Choy
AU - Ruixiang Wu
AU - Tongxin Li
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19585
UR - https://arxiv.org/abs/2610.19585
AB - In this work, we propose a curve to be a minimum of length among rectifiable, continuous curves of unit width in three-dimensional Euclidean space. We show that a previously conjectured minimizer of that problem is not of unit width, give the numerical derivation of a candidate family of unit width curves, prove that it admits a strict local minimum within that candidate family and that this minimum is of unit width and length 3.905124162243593…. The curve consists of eight straight segments, two cone geodesics, and three circular helices. Its parameters are specified by a uniquely isolated solution of the joining and stationarity equations. To establish unit width in every direction, we combine analytic estimates with interval arithmetic on a finite covering of the remaining directions. We conjecture that the curve is globally minimizing. We supplement this work with the programs used to identify the candidates, replicate the computer-assisted parts of our proof, as well as a formalization of our proof in Lean.
ER -