(i) State and prove the isoperimetric inequality for plane curves. You may appeal to Wirtinger's inequality as long as you state it precisely.

(ii) State Fenchel's theorem for curves in space.

(iii) Let $\alpha: I \rightarrow \mathbb{R}^{2}$ be a closed regular plane curve bounding a region $K$. Suppose $K \supset\left[p_{1}, p_{1}+d_{1}\right] \times\left[p_{2}, p_{2}+d_{2}\right]$, for $d_{1}>0, d_{2}>0$, i.e. $K$ contains a rectangle of dimensions $d_{1}, d_{2}$. Let $k(s)$ denote the signed curvature of $\alpha$ with respect to the inward pointing normal, where $\alpha$ is parametrised anticlockwise. Show that there exists an $s_{0} \in I$ such that $k\left(s_{0}\right) \leqslant \sqrt{\pi /\left(d_{1} d_{2}\right)}$.