  Let X be a compact complex manifold endowed with a big line bundle L. We
define the energy at equilibrium of a weighted subset as the Monge-Ampere
energy of the associated extremal plurisubharmonic weight. We prove the
differentiability of the energy at equilibrium with respect to the weight, and
show that this energy describes the asymptotic behaviour as k goes to infinity
of the volume of the induced sup-norm unit ball in the space of global sections
of kL. As a consequence of these results, we extend Rumely's Robin-type formula
for the transfinite diameter. We also obtain an asymptotic description of the
analytic torsion and extend Yuan's equidistribution theorem for algebraic
points of small height to the case of a big line bundle.
