Universal Lower Bounds for Potential Energy of Sph

Authors: P G Boyvalenkov P D Dragnev D P Hardin E B Saff M M Stoyanova

Publish Date: 2016/02/29

Volume: 44, Issue: 3, Pages: 385-415

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Abstract

We derive and investigate lower bounds for the potential energy of finite spherical point sets spherical codes Our bounds are optimal in the following sense—they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte–Yudin method However improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds All our bounds can be obtained in a unified manner that does not depend on the potential function provided the potential is given by an absolutely monotone function of the inner product between pairs of points and this is the reason we call them universal We also establish a criterion for a given code of dimension n and cardinality N not to be LPuniversally optimal eg we show that two codes conjectured by Ballinger et al to be universally optimal are not LPuniversally optimalFor the proof of Theorem 44b we show for each nge 3 and s in the interior of I 2k1 that Q 2k+3ns0 for k sufficiently large and thus by Theorem 41 the bound R 2k1nNh can be improved For this purpose we present several lemmas

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