### Abstract

The open problem of determining the exact value of the n-th linear polarization constant c_{n} of Rn has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup _{∥y∥=1}|〈x_{1}, y〉⋯〈x _{n}, y〉|, where x_{1},...,x_{n} are unit vectors in ℝ^{n}. The new estimate is given in terms of the eigenvalues of the Gram matrix [〈x_{i}, x_{j}〉] and improves upon earlier estimates of this kind. However, the intriguing conjecture c_{n} = n^{n/2} remains open.

Original language | English |
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Pages (from-to) | 304-310 |

Number of pages | 7 |

Journal | Linear Algebra and Its Applications |

Volume | 405 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Aug 1 2005 |

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### Keywords

- Gram matrices
- Linear polarization constants
- Polynomials over normed spaces
- Product of functionals

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

**A geometric estimate on the norm of product of functionals.** / Matolcsi, M.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 405, no. 1-3, pp. 304-310. https://doi.org/10.1016/j.laa.2005.03.028

}

TY - JOUR

T1 - A geometric estimate on the norm of product of functionals

AU - Matolcsi, M.

PY - 2005/8/1

Y1 - 2005/8/1

N2 - The open problem of determining the exact value of the n-th linear polarization constant cn of Rn has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup ∥y∥=1|〈x1, y〉⋯〈x n, y〉|, where x1,...,xn are unit vectors in ℝn. The new estimate is given in terms of the eigenvalues of the Gram matrix [〈xi, xj〉] and improves upon earlier estimates of this kind. However, the intriguing conjecture cn = nn/2 remains open.

AB - The open problem of determining the exact value of the n-th linear polarization constant cn of Rn has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup ∥y∥=1|〈x1, y〉⋯〈x n, y〉|, where x1,...,xn are unit vectors in ℝn. The new estimate is given in terms of the eigenvalues of the Gram matrix [〈xi, xj〉] and improves upon earlier estimates of this kind. However, the intriguing conjecture cn = nn/2 remains open.

KW - Gram matrices

KW - Linear polarization constants

KW - Polynomials over normed spaces

KW - Product of functionals

UR - http://www.scopus.com/inward/record.url?scp=21244441101&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21244441101&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2005.03.028

DO - 10.1016/j.laa.2005.03.028

M3 - Article

AN - SCOPUS:21244441101

VL - 405

SP - 304

EP - 310

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -