Idempotent tropical matrices and finite metric spaces

There is a well-known correspondence between the triangle inequality for a distance function on a finite set, and idempotency of an associated matrix over the tropical semiring. Recent research has shed new light on the structure (algebraic, combinatorial and geometric) of tropical idempotents, and in this paper we explore the consequences of this for the metric geometry of tropical polytopes. We prove, for example, that every n-point metric space is realised by the Hilbert projective metric on the tropical vertices of a pure n-dimensional, tropical and convex polytope in tropical n-space. More generally, every n-point asymmetric distance function is realised by a residuation operator on the vertices of such a polytope. In the symmetric case, we show that the maximal group of tropical matrices containing the idempotent associated to a metric space is isomorphic to G x R, where G is the isometry group of the space. From this we deduce that every group of the form GxR with G finite arises as a maximal subgroup of a sufficiently large finitary full tropical matrix semigroup. In the process we also prove some new results about tropical idempotent matrices, and note some semigroup-theoretic consequences which may be of independent interest.