  Motivated by conjectures in Mirror Symmetry, we continue the study of the
real Monge--Amp\`ere operator on the boundary of a simplex. This can be
formulated in terms of optimal transport, and we consider, more generally, the
problem of optimal transport between symmetric probability measures on the
boundary of a simplex and of the dual simplex. For suitably regular measures,
we obtain regularity properties of the transport map, and of its convex
potential. To do so, we exploit boundary regularity results for optimal
transport maps by Caffarelli, together with the symmetries of the simplex.
