  We study the Ginzburg-Landau functional describing an extreme type-II
superconductor wire with cross section with finitely many corners at the
boundary. We derive the ground state energy asymptotics up to $ o(1) $ errors
in the surface superconductivity regime, i.e., between the second and third
critical fields. We show that, compared to the case of smooth domains, each
corner provides an additional contribution of order $ \mathcal{O}(1) $
depending on the corner opening angle. The corner energy is in turn obtained
from an implicit model problem in an infinite wedge-like domain with fixed
magnetic field. We also prove that such an auxiliary problem is well-posed and
its ground state energy bounded and, finally, state a conjecture about its
explicit dependence on the opening angle of the sector.
