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
DOI:
https://doi.org/10.1090/S0002-9939-96-03087-0
MathSciNet review:
1301512
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Abstract | References | Similar Articles | Additional Information
Abstract: Consider the curve , where
is absolutely continuous on
. Then
has finite length, and if
is the
-neighborhood of
in the uniform norm, we compare the length of the shortest path in
with the length of
. Our main result establishes necessary and sufficient conditions on
such that the difference of these quantities is of order
as
. 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