### 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 n^{th} linear polarization constant C_{n}(ℂ^{n}) is n^{n/2}, for arbitrary n ∈ ℕ. The same value for c _{n}(ℝ^{n}) is only conjectured. In a recent work A. Pappas and S. Révész prove that c_{n}(ℝ^{n}) = n^{n/2} for n ≤ 5. Moreover, they show that if the linear forms are given as f_{j}(x) = 〈x, a_{j}〉, for some unit vectors a_{j} (1 ≤ j ≤ n), then the product of the f_{j}'s attains at least the value n^{-n/2} at the normalized signed sum of the vectors {a_{j}}_{j=1} having maximal length. Thus they asked whether this phenomenon remains true for arbitrary n ∈ ℕ. We show that for vector systems {a_{j}}_{j=1}^{n} 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 c_{n}(ℝ^{n}).

Original language | English |
---|---|

Pages (from-to) | 485-494 |

Number of pages | 10 |

Journal | Mathematical Inequalities and Applications |

Volume | 9 |

Issue number | 3 |

Publication status | Published - Jul 2006 |

### Fingerprint

### Keywords

- Linear polarizations constants
- Polynomial norm estimates
- Polynomials

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Inequalities and Applications*,

*9*(3), 485-494.

**On the real linear polarization constant problem.** / Matolcsi, M.; Muñoz, Gustavo A.

Research output: Contribution to journal › Article

*Mathematical Inequalities and Applications*, vol. 9, no. 3, pp. 485-494.

}

TY - JOUR

T1 - On the real linear polarization constant problem

AU - Matolcsi, M.

AU - Muñoz, Gustavo A.

PY - 2006/7

Y1 - 2006/7

N2 - 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).

AB - 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).

KW - Linear polarizations constants

KW - Polynomial norm estimates

KW - Polynomials

UR - http://www.scopus.com/inward/record.url?scp=33747078503&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747078503&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33747078503

VL - 9

SP - 485

EP - 494

JO - Mathematical Inequalities and Applications

JF - Mathematical Inequalities and Applications

SN - 1331-4343

IS - 3

ER -