Distances on the tropical line determined by two points

Abstract

Let p ′ , q ′ ∈ R n . Write p ′ ∼ q ′ if p ′ − q ′ is a multiple of (1, . . . , 1). Two different points p and q in R n / ∼ uniquely determine a tropical line L(p, q), passing through them, and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on n leaves. It is also a metric graph.If some representatives p ′ and q ′ of p and q are the first and second columns of some real normal idempotent order n matrix A, we prove that the tree L(p, q) is described by a matrix F , easily obtained from A. We also prove that L(p, q) is caterpillar. We prove that every vertex in L(p, q) belongs to the tropical linear segment joining p and q. A vertex, denoted pq, closest (w.r.t tropical distance) to p exists in L(p, q). Same for q. The distances between pairs of adjacent vertices in L(p, q) and the distances d(p, pq), d(qp, q) and d(p, q) are certain entries of the matrix |F |. In addition, if p and q are generic, then the tree L(p, q) is trivalent. The entries of F are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of A.

Isocanted alcoved polytopes

“…11 As a set, a tropical segment 12 is a finite union of classical segments. The tropical line strictly contains the tropical segment determined by two given points, and the difference set is a finite union of halflines; see [11,25,27]. For an alcoved polytope P, this implies that the skeleton 13…”

Quasi-Euclidean classification of alcoved convex polyhedra

“…We will use tropical distance 16 to measure edge-lengths: Tropical distance between integral points is just integer (i.e., lattice) distance. This distance (and a related seminorm and norm) have been used since 1979 in [5,16,17,18,19,20,28,29,39,40]. In the literature, they are called by various names, such as tropical Hilbert projective distance or Chebyshev distance and range seminorm.…”