Jordan Curve and Arc Length

Suppose $\alpha$ is a Jordan curve. Suppose $\beta$ is another Jordan curve such that $\alpha$ is contained in the region limited by $\beta$.

Maybe it is an easy question, but, how can one show that the length of $\beta$ is bigger than or equal to the length of $\alpha$?

Is the length of $\beta$ equal to length of $\alpha$ if and only if $\alpha=\beta$?

Thanks

You cannot. In fact, $\alpha$ can be as wiggly as you want, even having infinite length, while $\beta$ has finite length.
• Impressive, but is the statement true if the curve $\alpha$ has finite length? – Tomás Oct 22 '12 at 19:53
• No. And I can't imagine why you would believe it to be true. Try it yourself: Draw a circle on a sheet of paper, call it $\beta$. Inside it, draw a long, convoluted curve, wiggling about all over the place, taking care for the curve not to intersect itself. You can always continue the curve to end up at the starting point, thus closing the curve. Call it $\alpha$. Surely, if you put any effort into this, $\alpha$ will be way longer than $\beta$? – Harald Hanche-Olsen Oct 23 '12 at 6:17