  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.
