in the second inclusion of 5.

7.  Let $X,Y$ be closed. Then $X\subseteq Y$ if and only if ${\ensuremath{\mu_{X}}}\geq{\ensuremath{\mu_{Y}}}$.

\[MinkoExa\] As two simple examples, consider the $n$-simplex $\Delta^n\subseteq\mathbb R^n$ and a convex cone $C\subseteq\mathbb R^n$. Then (here $\geq$ denotes the product order on $\mathbb R^n$) $${\ensuremath{\mu_{\Delta^n}}}(x)= 
        \begin{cases}
            \sum_{j=1}^n\xi_j &\hspace*{-3mm},\quad x=(\xi_1,\ldots,\xi_n)\geq 0,
            \\
            \infty &\hspace*{-3mm},\quad \text{otherwise}.
        \end{cases}
        \qquad 
        {\ensuremath{\mu_{C}}}(x)=
        \begin{cases}
            0 &\hspace*{-3mm},\quad x\in C,
            \\
            \infty &\hspace*{-3mm},\quad \text{otherwise}.
        \end{cases}$$ Observe that $\Delta^n=\{x\in\mathbb R^n\mid {\ensuremath{\mu_{\Delta^n}}}(x)\leq 1\}$.

Another illustrative example is given by general pyramids in a euclidean space. This example will play an important role later on.

\[Pyramid\] For $u\in\mathbb R^n$ consider the set $$X=\big\{x\in\mathbb R^n\mid x\geq 0\text{ and }(x,u)\leq 1\big\}
        ,$$ where $(\cdot,\cdot)$ denotes the euclidean scalar product on $\mathbb R^n$. The set $X$ is intersection of the cone $\mathbb R_+^n$ with the half-space given by the inequality $(x,u)\leq 1$, hence it is convex and contains $0$. Thus $X$ is a .

Let us first assume that $u$ is strictly positive, i.e., $u\geq 0$ and no component of $u$ equals zero. Then $X$ is a pyramid (in $2$-dimensional space, a triangle).

(0,0) – (0,4-5/12) – (11-12/5,0) – (0,0); (0,0) – (12,0); (0,0) – (0,5); (-1,4) – (11,-1); (0,0) – (1,1\*12/5); at 