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Some remarks on the variation
of curve length and surface area


Authors: James Kuelbs and Wenbo V. Li
Journal: Proc. Amer. Math. Soc. 124 (1996), 859-867
MSC (1991): Primary 28A75, 41A29; Secondary 26A45, 49J40
MathSciNet review: 1301512
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Abstract | References | Similar Articles | Additional Information

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.


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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

DOI: https://doi.org/10.1090/S0002-9939-96-03087-0
Keywords: Shortest curve length, approximation with constraints, bounded variation, surfaces
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
Article copyright: © Copyright 1996 American Mathematical Society