$a_{i}>0$, $\forall i$.

2.  The set $C_{0}$ is a fundamental domain of $\mathcal{C}_{K}$ under the action of $D_{K}(1,n)$. In particular, $$\mathcal{C}_{K} = D_{K}(1,n)\cdot C_{0}$$

The reduction algorithm
-----------------------

Theorem \[thm:Diff+ReducedClasses\] and Corollary \[cor:FundamentalDomain\] lead to a simple algorithm to decide whether a given class in ${\mathcal{P}}_{+}$ belongs to $\mathcal{C}_{K}$. To simplify notation, let us write $(a_{0}\,;a_{1},\ldots,a_{n})$ for the class $a_{0}L-\sum_{i}a_{i}E_{i}$.

-   Set $\ell=-1$ and pick $v_{0}:=(a_{0}^{0}\,;a_{1}^{0},\ldots,a_{n}^{0})\in {\mathcal{P}}_{+}$.

-   Increment $\ell$ by one. If $a_{1}^{\ell}\geq\cdots\geq
      a_{n}^{\ell}$, set $\hat{v}_{\ell}=v_{\ell}$ and go to Step 3. Otherwise, using reflections $r_{\alpha_{i}}\in D_{K}(1,n)$ about $\alpha_{i}=E_{i}-E_{i+1}$, $1\leq i\leq n-1$, permute the coefficients of $v_{\ell}$ so that $a_{1}^{\ell}\geq a_{2}^{\ell}\geq\cdots\geq a_{n}^{\ell}$ and write $\hat{v}_{\ell}$ for the reordered vector.

-   Let $a_{n}^{\ell}$ be the last coefficient in $\hat{v}_{\ell}$. If $a_{n}^{\ell}<0$, then $v_{\ell}\not\in
      \mathcal{C}_{K}$. Thus $v_{0}\not\in\mathcal{C}_{K}$ and the algorithm stops.

-   Let $d_{\ell}:=a_{1}^{\ell}+a_{2}^{\ell}+a_{3}^{\ell}-a_{0}^{\ell}$. If $d_{\ell}\leq 0$, then the class $\hat{v}_{\ell}$ is reduced. In that case

    -   If $a_{n}^{\ell}>0$, then $\hat{v}_{\ell}\in\mathcal{C}_{K}$, hence $v_{0}\in\mathcal{C}_{K}$ as well, and the algorithm stops.

    -   If $a_{n}^{\ell}=0$, then $\hat{v}_{\ell}$ is in the boundary of $\mathcal{C}_{K}$, hence $v_{0}$ is in the boundary of $\mathcal{C}_{K}$ as well, and the algorithm stops.

-   The class $\hat{v}_{\ell}$ has non-negative coefficients but is not reduced. Apply the reflexion $r_{\alpha_{0}}\in
      D_{K}(1,n)$ about $\alpha_{0}=L-E_{1}-E_{2}-E_{3}$ 