Let $M$ be a matroid on the set $\{1,\dots,m\}$. The Bergman fan $\tilde{B}(M)$ is defined in literature to be the set in $\mathbb{R}^{m}$ consisting of the vectors $v=(w\_1,\dots,w\_m)$ such that for every circuit $c\in C(M)$, the minimum $\min\_{i\in c}w\_i$ is attained at least twice. 


It is then mostly left implicit what the fan structure on this set should be. Sometimes people mention equivalence classes formed by $v\_1\sim v\_1$ iff $M\_{v\_1}=M\_{v\_2}$, where $M\_v$ is the matroid with as bases those bases $B$ of $M$ attaining the minimum $\min\_{B\in B(M)} \sum\_{b\in B}v\_b$. For what I gather it is implied that these should define a fan structure on $\tilde{B}(M)$, but simple examples show that it doesn't:


Let $M=U\_{3,3}$. Then there are no circuits, so the support of $\tilde{B}(M)$ is the whole space. Since there is only a single basis, there is only a single equivalence class. So the equivalence classes do not define a fan structure on $\tilde{B}(M)$.


What am I not getting here?



Definition of the Bergman fan
From [Matroid polytopes, nested sets and Bergman fans](https://arxiv.org/abs/math/0411260v1), Feichtner and Sturmfels, just above Proposition 2.5 on the bottom of page 4:



> 
> Two vectors $w,w' \in \mathbb{R}^n$ are considered equivalent for the matroid $M$ if $M\_w = M\_{w'}$. The equivalence classes are relatively open convex polyhedral cones. These cones form a complete fan in $\mathbb{R}^n$. This fan is the normal fan of $P\_M$. If $\Gamma$ is a cone in the normal fan of $P\_M$ and $w \in \Gamma$ then we write $M\_\Gamma = M\_w$. The following proposition shows that the
>  Bergman fan $\tilde{\mathcal{B}}(M)$ is a subfan of the normal fan of the matroid polytope $P\_M$.
> 
> 
> 


For $M = U(3,3)$ the one equivalence class is $\mathbb{R}^3$, which is indeed open, convex and a cone (it is the positive hull of $\{\pm e\_1, \pm e\_2, \pm e\_3\}$). The set $\{\mathbb{R}^3\}$ is a fan:


* every face of a cone in the fan is in the fan (check: $\mathbb{R}^3$ is the only face)
* the intersection of any two cones in the fan is a face of each (check: $\mathbb{R}^3$ is the only intersection)




---


I tried working out another example but I'm still new to this so bear with me. For another example, let $M = U(1,3)$. Then the equivalence classes can be parameterized in terms of subsets of the bases of $M$:


* $M\_v$ has only $1$ as a basis iff $v = (a,b,b)$ with $a > b$.
* $M\_v$ has $1,2$ as bases iff $v = (a,a,b)$ with $a > b$
* $M\_v$ has $1,2,3$ as bases iff $v = (a,a,a)$


and we have similar conditions for permutations of the variables.


Therefore the equivalence classes are $$[1,0,0], [0,1,0], [0,0,1], [1,1,0], [1,0,1], [0,1,1], [1,1,1]$$ where $[1,0,0] = -[0,1,1], [0,1,0] = -[1,0,1], [0,0,1] = -[1,1,0]$. The line $[1,1,1]$ is a face of every cone. The intersecton of any two cones $C\_1, C\_2$ is either the line $[1,1,1]$ or if $C\_1 = C\_2$ then $C\_1 \cap C\_2 = C\_1$.


