Fermat–Weber points with respect to an asymmetric tropical distance function are studied. It
turns out that they correspond to the optimal solutions of a transportation problem. The …

In this paper we consider the categorical Isbell completion construction for generalized
metric spaces in the sense of Lawvere. We show that this is an analogue of the tight span …

Many important problems in extremal combinatorics can be stated as proving a pure
binomial inequality in graph homomorphism numbers, ie, proving that hom (H 1, G) a 1⋯ …

In 1970, Farris introduced a procedure that can be used to transform a tree metric into an
ultra metric. Since its discovery, Farris' procedure has been used extensively within …

The Gröbner fan of a polynomial ideal I⊆ k [x1,..., xn] is a polyhedral complex in Rn whose
maximal cones are in bijection with the reduced Gröbner bases of I. In tropical algebraic …

An algorithm to give an explicit description of all the solutions to any tropical linear system
A⊙ x= B⊙ x is presented. The given system is converted into a finite (rather small) number p …

Consider the space M n nor of square normal matrices X=(xij) over R∪{−∞}, ie,−∞≤ xij≤ 0
and xii= 0. Endow M n nor with the tropical sum⊕ and multiplication⊙. Fix a real matrix A∈ …

An extension (V, d) of a metric space (S, μ) is a metric space with S ⊆ V and d ∣ _S= μ, and
is said to be tight if there is no other extension (V, d′) of (S, μ) with d′≤ d. Isbell and Dress …

We show that in the constant coefficient case the generic tropical variety of a graded ideal
exists. This can be seen as an analogue to the existence of the generic initial ideal in …

The tight-span of a finite metric space is a polytopal complex with a structure that reflects
properties of the metric. In this paper we consider the tight-span of a totally split …