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I am staring at the proof of Lemma 37.5 in Lectures on Discrete and Polyhedral Geometry, see page 331. I cannot understand why the required triangulation exists. In the first paragraph it says "...

How can one find the volume of all Voronoi cells (bounded and unbounded) in an $n$-dimensional bounded space? For instance, consider an $N$-dimensional space (hypercube) with bounds on each dimension ...