Some remarks on the variation of curve length and surface area
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- by James Kuelbs and Wenbo V. Li PDF
- Proc. Amer. Math. Soc. 124 (1996), 859-867 Request permission
Abstract:
Consider the curve $C=\{ (t,f(t):0\le t\le 1\}$, where $f$ is absolutely continuous on $[0,1]$. Then $C$ has finite length, and if $U_{\epsilon }$ is the $\epsilon$-neighborhood of $f$ in the uniform norm, we compare the length of the shortest path in $U_{\epsilon }$ with the length of $f$. Our main result establishes necessary and sufficient conditions on $f$ such that the difference of these quantities is of order $\epsilon$ as $\epsilon \rightarrow 0$. We also include a result for surfaces.References
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Additional Information
- James Kuelbs
- Affiliation: Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin 53706
- Email: Kuelbs@math.wisc.edu
- Wenbo V. Li
- Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
- Email: Wli@math.udel.edu
- Received by editor(s): December 16, 1993
- Received by editor(s) in revised form: September 19, 1994
- Additional Notes: Supported in part by NSF grant number DMS-9024961.
- Communicated by: Andrew M. Bruckner
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 859-867
- MSC (1991): Primary 28A75, 41A29; Secondary 26A45, 49J40
- DOI: https://doi.org/10.1090/S0002-9939-96-03087-0
- MathSciNet review: 1301512