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?

Combinatorial convexity and algebraic geometryby Günter Ewald, to which Ziegler'sLectures on Polytopesrefers in his discussion of fans. Just a guess; I don't have Ewald's book. $\endgroup$ – Joseph O'Rourke Jul 11 '11 at 12:52