  A notion of weighted multivariate majorization is defined as a preorder on
sequences of vectors in Euclidean space induced by the Choquet ordering for
atomic probability measures. We characterize this preorder both in terms of
stochastic matrices and convex functions and use it to describe the
distribution of equilibrium points of logarithmic potentials generated by
discrete planar charge configurations. In the case of $n$ positive charges we
prove that the equilibrium points satisfy $\binom{n}{2}$ weighted majorization
relations and are uniquely determined by $n-1$ such relations. It is further
shown that the Hausdorff geometry of the equilibrium points and the charged
particles is controlled by the weighted standard deviation of the latter. By
using finite-rank perturbations of compact normal Hilbert space operators we
establish similar relations for infinite charge distributions. We also discuss
a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the
existence of zeros of Borel series with positive $l^1$-coefficients, and an
operator version of the Clunie-Eremenko-Rossi conjecture.
