Displaying the cohomology of toric line bundles

There is a standard approach to calculate the cohomology of torus-invariant sheaves $\mathcal{L}$ on a toric variety via the simplicial cohomology of the associated subsets $V(\mathcal{L})$ of the space $N_\mathbb{R}$ of 1-parameter subgroups of the torus. For a line bundle $\mathcal{L}$ represented by a formal difference $\Delta^+-\Delta^-$ of polyhedra in the character space $M_\mathbb{R}$, [1] contains a simpler formula for the cohomology of $\mathcal{L}$, replacing $V(\mathcal{L})$ by the set-theoretic difference $\Delta^- \setminus \Delta^+$. Here, we provide a short and direct proof of this formula.