Page 151 -

8 Optimal Transportation and Monge–Ampère Equations
146

Fig. 8.11. Favela in Rio de Janeiro. No urban planning whatsoever …

many collateral issues in urban planning are neglected by the model (like the
historic growth situation of cities, formation of low income and slum areas by
uncontrolled immigration into urban areas, city topography etc.). No existing
city has yet been planned by this model (and most likely never will be) but nev-
ertheless it serves as an interesting starting point for further modeling and as
an educational tool for city planners. In the Buttazzo–Santambrogio model the
transportation cost is accounted for by a Wasserstein distance of the densities
f and g , overcrowding is avoided by penalising with an ‘unhappiness’ func-
tional of f heavily penalizing population densities f , which are not absolutely
continuous with respect to the Lebesgue measure, and service concentration
is built in by heavily penalizing non-atomic service measures g.Then, atotal
cost functional is defined by summing up these three terms and minimizing
densities f and g are sought (with equal prescribed total mass). Note that a clas-
sical Monge–Kantorovich mass transportation problem appears here only as
a subproblem defining the transportation cost between residential and service
areas!
Typical minimizers (f , g) have the form of a certain number of circular resi-
dential areas with a service pole (atom of g) in the center. The radial population
density in these ‘subcities’ decreases away from the service pole.