  A remarkable connection between optimal design and Monge transport was
initiated in the years 1997 in the context of the minimal elastic compliance
problem and where the euclidean metric cost was naturally involved. In this
paper we present different variants in optimal design of mechanical structures,
in particular focusing on the optimal pre-stressed elastic membrane problem. We
show that the underlying metric cost is associated with an unknown maximal
monotone map which maximizes the Monge-Kantorovich distance between two
measures. In parallel with the classical duality theory leading to existence
and (in a smooth case) to PDE optimality conditions, we present a general
geometrical approach arising from a two-point scheme in which geodesics with
respect to the optimal metric play a central role. These two aspects are
enlightened by several explicit examples and also by numerical solutions in
which optimal structures very often turn out to be truss-like i.e supported by
piecewise affine geodesics. In case of a discrete load, we are able to relate
the existence of such truss-like solutions to an extension property of maximal
monotone maps which is of independent interest and that we propose here as a
conjecture.
