A Fractional Notion of Length and an Associated Nonlocal Curvature

  • Brian SeguinEmail author


In this paper, 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. 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.


\(\sigma \)-Length functional Nonlocal curvature-vector Fractional perimeter 

Mathematics Subject Classification

53A04 28A75 49Q15 



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Authors and Affiliations

  1. 1.Loyola University ChicagoChicagoUSA

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