Putnam 1950 B6

6. Consider the closed plane curves $C_{i}$ and $C_{o}$, their respective lengths $\left|C_{i}\right|$ and $\left|C_{o}\right|$, the closed surfaces $S_{i}$ and $S_{o}$, and their respective areas $\left|S_{i}\right|$ and $\left|S_{o}\right|$. Assume that $C_{i}$ lies inside $C_{o}$ and $S_{i}$ inside $S_{o}$. (Subscript $i$ stands for "inner," $o$ for "outer.") Prove the correct assertions among the following four, and disprove the others. (i) If $C_{i}$ is convex, $\left|C_{i}\right| \leq\left|C_{o}\right|$. (ii) If $S_{i}$ is convex, $\left|S_{i}\right| \leq\left|S_{o}\right|$. (iii) If $C_{o}$ is the smallest convex curve containing $C_{i}$, then $\left|C_{o}\right| \leq\left|C_{i}\right|$. (iv) If $S_{o}$ is the smallest convex surface containing $S_{i}$, then $\left|S_{o}\right| \leq\left|S_{i}\right|$.

You may assume that $C_{i}$ and $C_{o}$ are polygons and $S_{i}$ and $S_{o}$ polyhedra. (Why?)