If $f_1\colon X_1\to Y_1,f_2\colon X_2\to Y_2$ with $X_1\subseteq X_2,Y_1\subseteq Y_2$ and $f_2|_{X_1}=f_1$, then ${{\ensuremath{\widehat F}}}f_2|_{{{\ensuremath{\widehat F}}}X_1}={{\ensuremath{\widehat F}}}f_1$.

Note that [[$\widehat F$]{}]{} does not preserve direct products.

For a  $X$ whose carrier is a compact subset of a euclidean space, ${{\ensuremath{\widehat F}}}X$ can be described with help of a geometric notion, namely using the Minkowksi functional of $X$. Before we can state this fact, we have to make a brief digression to explain this notion and its properties.

\[Minko\] Let $X\subseteq\mathbb R^n$ be a . The *Minkowski functional* of $X$ is the map ${\ensuremath{\mu_{X}}}\colon \mathbb R^n\to[0,\infty]$ defined as ${\ensuremath{\mu_{X}}}(x)=\inf\{t>0\mid x\in tX\}$, where the infimum of the empty set is understood as $\infty$.

Minkowski functionals, sometimes also called *gauge*, are a central and exhaustively studied notion in convex geometry, see, e.g., [@rudin:1991 p.34] or [@rockafellar:1970 p.28].

We list some basic properties whose proof can be found in the mentioned textbooks.

1.  ${\ensuremath{\mu_{X}}}(px)=p{\ensuremath{\mu_{X}}}(x)$ for $x\in\mathbb R^n,p\geq 0$,

2.  ${\ensuremath{\mu_{X}}}(x+y)\leq {\ensuremath{\mu_{X}}}(x)+{\ensuremath{\mu_{X}}}(y)$ for $x,y\in\mathbb R^n$,

3.  ${\ensuremath{\mu_{X\cap Y}}}(x)=\max\{{\ensuremath{\mu_{X}}}(x),{\ensuremath{\mu_{Y}}}(x)\}$ for $x\in\mathbb R^n$.

4.  If $X$ is bounded, then ${\ensuremath{\mu_{X}}}(x)=0$ if and only if $x=0$.

The set $X$ can almost be recovered from ${\ensuremath{\mu_{X}}}$.

5.  ${\displaystyle\{x\in\mathbb R^n\mid{\ensuremath{\mu_{X}}}(x)<1\}\subseteq X\subseteq\{x\in\mathbb R^n\mid{\ensuremath{\mu_{X}}}(x)\leq 1\}}$.

6.  If $X$ is closed, equality holds 