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Geodesic distance Riesz energy on the sphere
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abstract
We study energy integrals and discrete energies on the sphere, in particular, analogs of the Riesz energy with the geodesic distance in place of Euclidean, and observe that the range of exponents for which the uniform distribution optimizes such energies is different from the classical case. We also obtain a general form of the Stolarsky principle, which relates discrete energies to certain $L^2$ discrepancies. This leads to new proofs of discrepancy estimates, as well as the sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case.
Forward citations
Cited by 2 Pith papers
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Optimal measures for p-frame energies on spheres
Tight designs minimize p-frame energies over all probability measures for p between consecutive even integers, and the 600-cell does so on S3 for p in [8,10].
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Energy on spheres and discreteness of minimizing measures
For non-even p, every minimizer of the p-frame energy on the sphere has support with empty interior, and for potentials with finitely many positive Gegenbauer coefficients a discrete minimizer always exists.
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