Tropical convexity in location problems

TL;DR

This paper explores tropical convexity in location problems, establishing theoretical foundations and applications in phylogenetics, including properties of consensus methods based on tropical quasiconvex functions.

Contribution

It introduces the class of tropically quasiconvex functions, linking tropical convexity with location optimization and phylogenetic consensus methods.

Findings

Optima lie in tropical convex hulls of input points.

Tropically quasiconvex functions relate to monotonic functions.

Applications to phylogenetics and consensus methods.

Abstract

We investigate location problems whose optimum lies in the tropical convex hull of the input points. Firstly, we study geodesically star-convex sets under the asymmetric tropical distance and introduce the class of tropically quasiconvex functions whose sub-level sets have this shape. The latter are related to monotonic functions. Then we show that location problems whose distances are measured by tropically quasiconvex functions as before give an optimum in the tropical convex hull of the input points. We also show that a similar result holds if we replace the input points by tropically convex sets. Finally, we focus on applications to phylogenetics presenting properties of consensus methods arising from our class of location problems.