# Reference for this polyhedral lemma

Recall the definition of a fan: Let $U$ be a finite dimensional real vector space. Then a fan is a collection $\mathcal{F}$ of cones in $U$ such that

(1) If $\sigma \in \mathcal{F}$ and $\tau$ is a face of $\sigma$, then $\tau \in \mathcal{F}$.

(2) If $\sigma_1$ and $\sigma_2 \in \mathcal{F}$, then $\sigma_1 \cap \sigma_2$ is a face of both $\sigma_1$ and $\sigma_2$.

The following is widely known: Let $\mathcal{S}$ be a collection of cones in $U$ obeying (2). Let $\mathcal{F}$ be the collection of all faces of cones in $\mathcal{S}$ (so $\mathcal{F}$ obeys (1)). Then $\mathcal{F}$ obeys (2).

Is there a standard person and reference to whom this observation is attributed?

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Added a bounty as an excuse for bumping this to the front page. I'm finishing up a paper which uses this fact, and I'd like to get my references right. –  David Speyer Jul 11 '11 at 12:33
You might look in Combinatorial convexity and algebraic geometry by Günter Ewald, to which Ziegler's Lectures on Polytopes refers in his discussion of fans. Just a guess; I don't have Ewald's book. –  Joseph O'Rourke Jul 11 '11 at 12:52
Grünbaum's book (Convex Polytopes, 2nd ed) has some discussion in 3.2: a fan is a special case of a polyhedral complex, for which notion he refers to the books of Alexandroff-Hopf and Lefschetz. I see Ewald has a similar discussion. Neither explicitly states what you want, it seems. They both observe that the boundary complex of a polyhedron (say, cone) is a polyhedral complex (fan). Grünbaum's exercise 7 makes you prove almost what you want. So you might try Alexandroff's Combinatorial Topology for an explicit reference, since it's at least in translation. –  Graham Denham Jul 11 '11 at 15:13