What is the rigorous definition of polyhedral fan? What are some good resources to learn about them? What context do they arise naturally in? 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?
 A: 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.
