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I've been reading about tropical geometry and many papers reference polyhedral fans. I feel like I have a decent intuitive picture of what they are from reading articles but I still haven't been able to guess the general definition. All the ones I've encountered have been systems of linear inequalities, so that is my best guess at a general definition.

Any comments on where they appeared first historically or links/books to general resources on learning about them would be appreciated. Also, I'm curious to know what other areas of math these show up in?

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A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A fan is a finite set $F$ of polyhedral cones, all living in the same vector space, such that

(1) if $\sigma$ is a cone in $F$, and $\tau$ is a face of $\sigma$, then $\tau$ is in $F$.

(2) if $\sigma$ and $\sigma'$ are in $F$, then $\sigma \cap \sigma'$ is a face of both $\sigma$ and of $\sigma'$.

This blog post of mine might help you visualize these definitions.

Most mathematicians I know learned fans from Fulton's Toric Varieties. This would involve learning a lot of algebraic geometry on top of your combinatorics, although it is algebraic geometry that is very relevant to tropical geometry.

For a pure combinatorics reference, have you tried Chapter 2 of De Loera, Rambau and Santos? They focus on polyhedral complexes, which is the more general setup where you don't require that the half spaces pass through $0$, but they talk about fans as well. I haven't had a chance to look at it yet but, based on my knowledge of the authors, I expect it is very good.

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This sounds a lot like a definition in Enumerative Combinatorics, around where Stanley talks about Ehrhart polynomials, but I don't think Stanley uses the word "fan." Hmm. –  Qiaochu Yuan Aug 8 '10 at 21:29
    
For Toric Varieties, rather than Fulton, I'd recommend Cox, Little and Schenck's book in progress available on David Cox's website (for now) –  Charles Siegel Aug 8 '10 at 22:24
    
Thanks for all the recommendations and explanations! –  WWright Aug 8 '10 at 23:10

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