# Extreme rays of the $\ell^\infty$-nearest ultrametric tropical polytope

## TL;DR

This paper investigates the structure of ultrametrics closest to a given dissimilarity map in the $\, ext{l}^\, ext{infty}$ norm, revealing that Bernstein's extremality condition is only sufficient for three nodes, not for four or more.

## Contribution

It demonstrates the limitations of Bernstein's necessary condition for extreme rays in the ultrametric tropical polytope for larger node sets and provides explicit constructions.

## Key findings

- Bernstein's condition is sufficient only for n=3
- Counterexamples show insufficiency for n≥4
- Explicit characterization of extreme rays for small n

## Abstract

The set of ultrametrics on $[n]$ nodes that are $\ell^\infty$-nearest to a given dissimilarity map forms a $(\max,+)$ tropical polytope. Previous work of Bernstein has given a superset of the set containing all the phylogenetic trees that are extreme rays of this polytope. In this paper, we show that Bernstein's necessary condition of tropical extreme rays is sufficient only for $n=3$ but not for $n\geq 4$. Our proof relies on the exterior description of this tropical polytope, together with the tangent hypergraph techniques for extremality characterization. The sufficiency of the case $n=3$ is proved by explicitly finding all extreme rays through the exterior description. Meanwhile, an inductive construction of counterexamples is given to show the insufficiency for $n\geq 4$.