We study energy integrals and discrete energies on the sphere, in particular, analogues of the Riesz energy with the geodesic distance in place of the Euclidean, and we determine that the range of exponents for which uniform distribution optimizes such energies is different from the classical case. We also obtain a very general form of the Stolarsky principle, which relates discrete energies to certain L2 discrepancies, and prove optimal asymptotic estimates for both objects. This leads to sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case, as well as new proofs of discrepancy estimates.
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Received by the editors November 9, 2017, and, in revised form, May 20, 2018, and July 1, 2018. 2010 Mathematics Subject Classification. Primary 11K38, 74G65; Secondary 42C10, 33C55. The stay of the first author at CRM (Barcelona) has been sponsored by NSF grant DMS 1613790. The work of the first author is partially supported by the Simons Foundation collaboration grant and NSF grant DMS 1665007. The work of the second author is partially supported by NSERC Canada under grant RGPIN 04702.