Some remarks on the variation of curve length and surface area

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.