ScalingStacks

Corollary 7.7 . [02X6]

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Corollary 7.7.

Let Δ⊂ℝn\Delta\subset\mathbb{R}^{n} be a polytope of dimension nn and u∈ℝnu\in\mathbb{R}^{n}. Then,

voln⁡(Δ)=∑V∈Δ⁡(u)∑k=0dim(V)Ck​(Δ,u,V)​⟨u,V⟩n−k(n−k)!.\operatorname{vol}_{n}(\Delta)=\sum_{V\in\Delta(u)}\sum_{k=0}^{\dim(V)}C_{k}(\Delta,u,V)\frac{\langle u,V\rangle^{n-k}}{(n-k)!}.

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