Positive Definiteness and the Stolarsky Invariance Principle

Ryan William Matzke, Dmitriy Bilyk, Oleksandr Vlasiuk

Research output: Contribution to journalArticlepeer-review


In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. In particular, assuming the existence of a K-invariant measure μ with full support, we show that conditional positive definiteness of a kernel K is equivalent to a long list of other properties: including, among others, convexity of the energy functional, inequalities for mixed energies, and the fact that μ minimizes the energy integral in various senses. In addition, we prove a very general form of the Stolarsky Invariance Principle on compact spaces, which connects energy minimization and discrepancy and extends several previously known versions.
Original languageEnglish
Article number126220
JournalJournal of Mathematical Analysis and Applications
Issue number2
Publication statusPublished - 2022


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