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Take a discrete rank $n$ free abelian subgroup $\Gamma$ of the group of diagonal matrices in $SL(n+1, \mathbb R)$ with positive diagonal entries. The group $\Gamma$ will preserve each open coordinate orthant $C_i, i=1,...,2^{n+1}$ in $\mathbb R^{n+1}$. The projectivization $P(C_i)\subset \mathbb R P^n$ of each $C_i$ is open and properly convex: There will be $2^n$ of these domains in $\mathbb R P^n$. The group $\Gamma$ preserves each $P(C_i)$, acts on it properly discontinuously and cocompactly.