A subset $\Pi\subset\mathbb{R}^{n+1}$ is called a proper rational polyhedral complex (or just a polyhedral complex in this paper) if it can be presented as a finite union of closed sets in $\mathbb{R}^{n+1}$ called cells with the following properties.

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  Each cell is a closed convex (possibly semi-infinite) polyhedron. The dimension of the cell is, by definition, the dimension of its affine spun, the smallest affine subspace of $\mathbb{R}^{n+1}$ which contains it. We call a cell of dimension $k$ a $k$-cell.

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  The slope of the affine spun of each cell is rational. I.e. the linear subspace of $\mathbb{R}^{n+1}$ parallel to the affine spun is defined over $\mathbb{Q}$.

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  The boundary (i.e. the boundary in the corresponding affine spun) of a $k$-cell is a union of $(k-1)$-cells.

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  Different open cells (i.e. the interiors of the cells in the corresponding affine spuns) do not intersect.