## Distances on the tropical line determined by two points.(English)Zbl 1321.14050

Let $$n\geq 3$$. Consider the tropical projective torus $$Q^{n-1}=\mathbb R^n/\mathbb R(1,\dots, 1)$$, endowed with the tropical distance $$d(p,q)=\max |p_i-q_i-p_j+q_j|$$. Let $$L$$ be the unique continuous map that assigns to each pair $$(p,q)$$ of distinct points in $$Q^{n-1}$$ a tropical line $$L(p,q)$$ passing through them. This line is a metric tree on $$n$$ leaves. It contains the tropical segment $\mathrm{tconv}(p,q)=\{\lambda \odot p\oplus \mu\odot q\quad|\quad\lambda, \mu\in\mathbb R\}.$
The paper under review describes the line $$L(p,q)$$ if $$p$$ and $$q$$ are (represented by) the first two columns of some real, normal, tropically idempotent $$n$$-square matrix, where ‘normal’ means that all diagonal entries are zero and all entries are non-positive. (Tropical addition is defined to be maximum, not minimum.) It is proved that every vertex of $$L(p,q)$$, excepting the leaves, belongs to the tropical linear segment joining $$p$$ and $$q$$, which may fail under less restrictive conditions. Moreover, $$L(p,q)$$ is a caterpillar tree, i.e., it contains a path passing through all vertices except the leaves. The distances between adjacent vertices, and also those between $$p$$ (resp. $$q$$) and the nearest vertex, are of the form $$|p_i-q_i-p_j+q_j|$$. In addition, if $$p$$ and $$q$$ are generic, then the tree $$L(p,q)$$ is trivalent.

### MSC:

 14T05 Tropical geometry (MSC2010) 15A80 Max-plus and related algebras
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### References:

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