MathDB

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)

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.

Problem Statement