  For any Borel probability measure on $\mathbb{R}^n$, we may define a family
of eccentricity tensors. This new notion, together with a tensorization trick,
allows us to prove an energy minimization property for rotationally invariant
probability measures. We use this theory to give a new proof of the Welch
bounds, and to improve upon them for collections of real vectors. In addition,
we are able to give elementary proofs for two theorems characterizing
probability measures optimizing one-parameter families of energy integrals on
the sphere. We are also able to explain why a phase transition occurs for
optimizers of these two families.
