Riesz polarization inequalities in higher dimensions

• Autores: Tamás Erdélyi, E. B. Saff
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 171, Nº 1, 2013, págs. 128-147
• Idioma: inglés
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• Resumen

  • We derive bounds and asymptotics for the maximum Riesz polarization quantity Mp n (A) := max x1,x2,...,xn�¸A min x�¸A .n j=1 1 | x . x j |p (which is n times the Chebyshev constant) for quite general sets A �¼ Rm with special focus on the unit sphere and unit ball.We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results.We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when A is the unit circle and p > 0, as well as provide an independent proof of their result for p = 4 that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.