  We study the geometry of tropical Fermat-Weber points in terms of the
symmetric tropical metric over the tropical projective torus. It is well known
that a tropical Fermat-Weber point of a given sample is not unique and in this
paper we show that the set of all possible Fermat-Weber points forms a
polytrope. Then, we introduce the tropical Fermat-Weber gradient and using
them, we show that the tropical Fermat-Weber polytrope is a bounded cell of a
tropical hyperplane arrangement given by both min- and max-tropical hyperplanes
with apices which are observations in the input data.
