Polarization, sign sequences and isotropic vector systems

We determine the order of magnitude of the $n$ -th $ℓ_{p}$ -polarization constant of the unit sphere $S^{d - 1}$ for every $n, d \geq 1$ and $p > 0$ . For $p = 2$ , we prove that extremizers are isotropic vector sets, whereas for $p = 1$ , we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for $d = 2$ , we discuss the optimality of equally spaced configurations on the unit circle.

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Keywords

polarization problems, discrete potentials, Chebyshev constants, isotropic vectors sets, tight frames, vector sums

Mathematical Subject Classification 2010

Primary: 52A40

Secondary: 41A17

Milestones

Received: 23 April 2019

Revised: 19 June 2019

Accepted: 19 June 2019

Published: 4 January 2020

Authors

Gergely Ambrus
Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest Hungary

Sloan Nietert
Department of Computer Science Cornell University Ithaca, NY United States

Vol. 303, No. 2, 2019

Gergely Ambrus and Sloan Nietert

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors