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Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow X_{i_1,...,i_m}$ and $$X_{i_1,...,i_m}= \mathrm min \frac {1}{2}( X_{i_1, i_{\sigma(1)}}+ X_{ i_{\sigma(1)},i_{\sigma^2(1) }} + \ldots X_{ i_{\sigma^{m-1}(1)},i_{\sigma^m(1) }} )$$ where $\sigma$ represents a cyclic permutation. So, basically $\phi^m$ takes a dissimilarity vector of a tree to a m-dissimilarity vector of that tree.

For what values of $m \gt 3$ and $n$ is it known that $$\phi^m (Trop(Gr(2,n)) = Trop(Gr(m,n)) \cap \phi^m (\mathbb R^{n \choose 2})$$ ? For what values it is known they can't be equal?

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I don't know the answer, but why not contact Filip Cools? He proved the equality when m=3, right? –  quim Apr 16 '13 at 8:42
    
@quim Yes, he did it. –  john Apr 16 '13 at 10:25
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