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8 Optimal Transportation and Monge–Ampère Equations
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Fig. 8.8. Urban planning in Buenos Aires, for the living …

wise smooth curve segments). Associated transportation plans are measures
representing the mass transported THROUGH a given traffic plan, connecting
the irrigating (source) measure with the irrigated (demand) measure. The cost
functional of traffic plan is set up according to infrastructural expenses and
constraints. In analogy to the Kantorovich cost functional a minimisation over
irrigation plans connecting a given supply to a given demand measure or, resp.,
over traffic plans with a given transportation plan is carried out, leading to the
simultaneous construction of the transport paths and the transportation plan.
Note that a major difference to the Kantorovich problem lies in the fact that the
cost function generally depends on the whole transportation path and not only
on its endpoints!
The Images 8.8–8.11 depict urban areas in Buenos Aires, Quito and Rio de
Janeiro. Recently, urban planning models based on Monge–Kantorovich mass
transportation have been introduced in the literature. We cite [4], where a very
interesting model of optimal distribution of residential areas (density f )and
service areas (density g)inanurban environmentispresented,which we shall
discuss in some detail thereafter. The model is based on the propositions that
there is a transportation cost for moving between residential and service areas,
that overcrowding of residential areas is typically avoided by city dwellers and
that concentration of services is desirable for increasing efficiency. Obviously,