[1808.08654] A fractional notion of length and an associated nonlocal curvature Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:1808.08654 (math) [Submitted on 27 Aug 2018 ( v1 ), last revised 27 Apr 2026 (this version, v6)] Title: A fractional notion of length and an associated nonlocal curvature Authors: Brian Seguin View a PDF of the paper titled A fractional notion of length and an associated nonlocal curvature, by Brian Seguin View PDF HTML (experimental) Abstract: Here a new notion of fractional length of a smooth curve, which depends on a parameter $\sigma$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant. Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve. This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature. Comments: 32 pages, 4 figures Subjects: Differential Geometry (math.DG) MSC classes: 53A04, 28A75, 49Q15 Cite as: arXiv:1808.08654 [math.DG] (or arXiv:1808.08654v6 [math.DG] for this version) https://doi.org/10.48550/arXiv.1808.08654 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Brian Seguin [ view email ] [v1] Mon, 27 Aug 2018 01:24:27 UTC (2,616 KB) [v2] Thu, 27 Sep 2018 01:30:43 UTC (2,616 KB) [v3] Mon, 24 Jan 2022 02:55:33 UTC (2,619 KB) [v4] Sun, 29 Jan 2023 21:56:35 UTC (2,619 KB) [v5] Sun, 19 Mar 2023 22:57:10 UTC (2,620 KB) [v6] Mon, 27 Apr 2026 22:27:58 UTC (3,843 KB) Full-text links: Access Paper: View a PDF of the paper titled A fractional notion of length and an associated nonlocal curvature, by Brian Seguin View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2018-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from