Volumes of cones

I was talking to a friend today who does some tutoring in secondary school maths in the UK (roughly the same as the beginning of high school in the US) who pointed out that the volume of a 2- or 3-dimensional cone of height 1 over a line or polygon is respectively 1/2 or 1/3 times the volume of the base. He wondered if the same result held in higher dimensions with the dimension of the cone (equivalently, one more than the dimension of the base) as denominator. In other words, if \mathcal{C}P denotes a cone of height 1 over a polytope P of dimension d, then Vol(\mathcal{C}P)=\frac{1}{d+1}Vol(P). This is indeed true, as may be known to some of you. As a Saturday afternoon diversion, I wrote two proofs of this equality: one direct proof after reducing to the case of simplices, and one using Ehrhart theory. The latter crops up regularly at the combinatorial end of algebraic geometry – naturally in toric geometry – where certain generating functions for lattice point counts are Hilbert series for toric varieties. You could try to recast the equality above in these terms if you were so inclined or desirous of diversion…

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