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Consider a polytope $P$ defined by the usual inequalities $A\mathbf{x}\leq \mathbf{b}$; let me assume that $P$ is not contained in a proper subspace. A result which I believe to true, but am not having much luck actually finding a reference for is:

The volume of $P$ varies piecewise polynomially with respect to $\mathbf{b}$.

I think I see the proof using mixed volumes or by decomposing $P$ into simplices (where the result is simple linear algebra), and presumably am just not finding the right words to Google, or this seems too obvious to mention for combinatorialists.

I have one slightly more specific result in mind that depends on this one; again, I am confidentish that I see the proof (just prove it for simplices), but it's really a tangent for the application where I want to use it and would much prefer to have a reference. Now assume that I vary $\mathbf{b}=\mathbf{b}_0+t\mathbf{b}_1$ linearly such that at $t\neq 0$, $\mathrm{Vol}(P_t)\neq 0$.

Theorem: The order of vanishing of $\mathrm{Vol}(P_t)$ at $t=0$ is equal to the codimension of the span of $P_0$ (i.e. the number of independent linear equations it satisfies).

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  • $\begingroup$ "Multivariate Ehrhart theory" might be a phrase to google. $\endgroup$
    – Sam Hopkins
    Commented May 30, 2022 at 15:08
  • $\begingroup$ @SamHopkins It's a good suggestion, but I do fear that the people working on Erhart theory think of the points above as too obvious to discuss. $\endgroup$
    – Ben Webster ♦
    Commented May 30, 2022 at 15:17
  • $\begingroup$ For example, see Theorem 2 of link.springer.com/content/pdf/10.1007/s00591-005-0094-9.pdf. $\endgroup$
    – Sam Hopkins
    Commented May 30, 2022 at 15:22
  • $\begingroup$ If A is integral you could of course blame the first result on Duistermaat-Heckman. $\endgroup$
    – Allen Knutson
    Commented May 30, 2022 at 15:35
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    $\begingroup$ @AllenKnutson That really seems like overkill. If there's another paper that proves this with DH, I'm happy to cite it, but in the context I'm writing, that's a very big tangent. $\endgroup$
    – Ben Webster ♦
    Commented May 30, 2022 at 15:42

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