Some remarks on the variation of curve length and surface area
Some remarks on the variation of curve length and surface area
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 …