IMC 1996 · Problem 6

Upper content of a subset $E$ of the plane $\mathbb{R}^2$ is defined as

$$\mathcal{C}(E) = \inf \left\{ \sum_{i=1}^{n} \operatorname{diam}(E_i) \right\}$$

where $\inf$ is taken over all finite families of sets $E_1, \dots, E_n$, $n \in \mathbb{N}$, in $\mathbb{R}^2$ such that $E \subset \bigcup\limits_{i=1}^{n} E_i$.

Lower content of $E$ is defined as

$$\mathcal{K}(E) = \sup \{ \operatorname{lenght}(L) \;:\; L \text{ is a closed line segment onto which } E \text{ can be contracted} \}.$$

Show that

(a) $\mathcal{C}(L) = \operatorname{lenght}(L)$ if $L$ is a closed line segment;

(b) $\mathcal{C}(E) \ge \mathcal{K}(E)$;

(c) the equality in (b) needs not hold even if $E$ is compact.