On the transportation cost norm on finite metric graphs

For a finite metric graph X=(V,E,\ell) , where V is endowed with the shortest path metric, we consider the transportation cost problem associated with the distance d on V . Namely, for f a function with total sum 0 on V , write f=\sum_{a,b\in V}P(a,b)(\delta_{a}-\delta_{b}) , where the transportation plan P satisfies P(a,b)\geq 0 for (a,b)\in V\times V . The cost of P is W(P):=\sum_{a,b\in V}P(a,b)d(a,b) and the transportation norm of f is \|f\|_{TC}=\min_{P} W(P) , where P runs over all transportation plans for f .In this semi-survey paper, we give short proofs for the following statements: We use this to reprove known formulae for the transportation norm when X is either a tree or a cycle.