  We establish lower semi-continuity and strict convexity of the energy
functionals for a large class of vector equilibrium problems in logarithmic
potential theory. This in particular implies the existence and uniqueness of a
minimizer for such vector equilibrium problems. Our work extends earlier
results in that we allow unbounded supports without having strongly confining
external fields. To deal with the possible noncompactness of supports, we map
the vector equilibrium problem onto the Riemann sphere and our results follow
from a study of vector equilibrium problems on compacts in higher dimensions.
Our results cover a number of cases that were recently considered in random
matrix theory and for which the existence of a minimizer was not clearly
established yet.
