Positive definiteness and the Stolarsky invariance principle

In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. particular, assuming existence of a K-invariant measure μ with full support, show that conditional definiteness kernel K is equivalent to long list other properties: including, among others, convexity functional, inequalities for mixed energies, fact minimizes integral in various senses. addition, prove very general form Stolarsky Invariance Principle compact spaces, which connects minimization discrepancy extends several previously known versions.