Distances on the tropical line determined by two points (Q2925642)

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 NEWLINE\[NEWLINE\mathrm{tconv}(p,q)=\{\lambda \odot p\oplus \mu\odot q\quad|\quad\lambda, \mu\in\mathbb R\}.NEWLINE\]NEWLINENEWLINENEWLINEThe 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.