A mass reducing flow for real-valued flat chains with applications to transport networks^*

Hendrix College, 1600 Washington Avenue, Conway, AR 72032-3080, USA.

^** Corresponding author: downes@hendrix.edu

Received: 26 May 2020
Accepted: 28 June 2021

Abstract

An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accommodate efficiencies of scale into the model, one uses a suitable M^α norm for transportation cost for α ∈ (0, 1]. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this paper, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this M^α mass reducing flow for real-valued flat chains by finding a higher dimensional real chain whose slices dictate the flow. Keeping the boundary fixed, this flow reduces the M^α mass of the initial chain and is Lipschitz continuous under the flat-α norm. To complete the paper, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient.