(9.5,1.7) [[$X$]{}]{}; at (14.8,5.0) [[$\scriptstyle(x,u)=1$]{}]{};

Now we return to the functor [[$\widehat F$]{}]{}.

\[Ghat-Minko\] Let $X\subseteq\mathbb R^n$ be a , and assume that $X$ is compact. Then $${{\ensuremath{\widehat F}}}X=\Big\{(o,\phi)\in\mathbb R\times(\mathbb R^n)^A\mid\ 
        o\geq 0,\ o+\sum_{a\in A}{\ensuremath{\mu_{X}}}(\phi(a))\leq 1 \Big\}
        .$$

In the following we use the elementary fact that every convex map has a linear extension.

\[LinExt\] Let $V_1,V_2$ be vector spaces, let $X\subseteq V_1$ be a , and let $c\colon X\to V_2$ be a convex map. Then $c$ has a linear extension $\tilde c\colon V_1\to V_2$. If $\operatorname{span}X=V_1$, this extension is unique.

Rescaling in this representation of ${{\ensuremath{\widehat F}}}X$ leads to a characterisation of [[$\widehat F$]{}]{}-coalgebra maps. We give a slightly more general statement; for the just said, use $X=Y$.

\[Ghat-Minko-Coalg\] Let $X,Y\subseteq\mathbb R^n$ be s, and assume that $X$ and $Y$ are compact. Further, let $c\colon X\to\mathbb R_+\times(\mathbb R^n)^A$ be a convex map, and let $\tilde c\colon \mathbb R^n\to\mathbb R\times(\mathbb R^n)^A$ be a linear extension of $c$.

Then $c(X)\subseteq{{\ensuremath{\widehat F}}}Y$, if and only if $$\label{2}
        {{\tilde c}_{\text{\rm o}}}(x)+\sum_{a\in A}{\ensuremath{\mu_{Y}}}({{\tilde c}_{a}}(x))\leq{\ensuremath{\mu_{X}}}(x),\quad x\in\mathbb R^n
        .$$

An extension theorem for ${{\ensuremath{\widehat F}}}$-coalgebras {#sec-ExtensionTheorem}
=================================================================

In this section we establish an extension theorem for ${{\ensuremath{\widehat F}}}$-coalgebras. It states that an ${{\ensuremath{\widehat F}}}$-coalgebra, whose carrier 