Tropical Fermat--Weber Problems over Non-Finite Data and their Inverse Formulations

Abstract

The term tropical pseudonorm refers to a family of (not necessarily symmetric) gauge functions that arise in tropical or idempotent geometry. An important characteristic of these gauges is their invariance under translation by a constant vector, allowing them to descent naturally to tropical projective spaces. In this work, we explore the tropical one-infinity pseudonorm, a polyhedral hybrid gauge that allows for tunable asymmetry, in the context of a Fermat--Weber location problem. We extend previous formulations in considering non-finite data, and we investigate several variants of the inverse problem, providing linear programming formulations for their solution.