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

Energy optimization for distributions on the sphere and improvement to the Welch bounds