IMC 1996 Problem 6

Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as $$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$ 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 $E\subset \bigcup_{i=1}^{n}E_{i}$ . Lower content of $E$ is defined as $$\mathcal{K}(E)=\sup\{\text{length}(L) | L \text{ is a closed line segment onto which $E$ can be contracted}\}$$ . Prove that i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment; ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$ ; iii) the equality in ii) is not always true even if $E$ is compact.