Plots

Below are two graphs of the potential of the MA metric on the standard unit simplex, plotted in a coordinate chart covering the star of a vertex. The potential is smooth near the origin, but a "crease" singularity develops as we pass over the singular points halfway along each edge, thus the location of the singular points in the affine structure is dictated by the PDE in the spirit of a free boundary problem. Computed on a 200 x 200 grid, using entropic regularisation of optimal transport.

The same potential plotted over slices perpendicular to one of the edges. Below the red circle, the curves are smooth, above it they are singular.