On the real linear polarization constant problem

Máté Matolcsi, Gustavo A. Muñoz

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna [2], that the so-called nth linear polarization constant Cn(ℂn) is nn/2, for arbitrary n ∈ ℕ. The same value for c n(ℝn) is only conjectured. In a recent work A. Pappas and S. Révész prove that cn(ℝn) = nn/2 for n ≤ 5. Moreover, they show that if the linear forms are given as fj(x) = 〈x, aj〉, for some unit vectors aj (1 ≤ j ≤ n), then the product of the fj's attains at least the value n-n/2 at the normalized signed sum of the vectors {aj}j=1 having maximal length. Thus they asked whether this phenomenon remains true for arbitrary n ∈ ℕ. We show that for vector systems {aj}j=1n close to an orthonormal system, the Pappas-Révész estimate does hold true. Furthermore, among these vector systems the only system giving n-n/2 as the norm of the product is the orthonormal system. On the other hand, for arbitrary vector systems we answer the question of A. Pappas and S. Révész in the negative when n ∈ ℕ is large enough. We also discuss various further examples and counterexamples that may be instructive for further research towards the determination of cn(ℝn).

Original languageEnglish
Pages (from-to)485-494
Number of pages10
JournalMathematical Inequalities and Applications
Volume9
Issue number3
DOIs
Publication statusPublished - Jul 2006

Keywords

  • Linear polarizations constants
  • Polynomial norm estimates
  • Polynomials

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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