The linear polarization constant of Rn

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The dth linear polarization constant of Rd

In [C. Benítez, Y. Sarantopoulos, A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (3) (1998) 395–408] it was conjectured that for all unit vectors u1, . . . , ud in Rd , X (u1, . . . , ud ) := sup x∈Rd , |x|2=d d ∏ i=1 〈x,ui〉 1 with equality occurring iff u1, . . . , ud are orthonormal. We relate this to a conjecture about solutions of Ay = y−...

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On the Linear Polarization Constant Of

The present work contributes to the determination of the n-th linear polarization constant cn(H) of an n-dimensional real Hilbert space H . We provide some new lower bounds on the value of sup‖y‖=1 | 〈x1, y〉 · · · 〈xn, y〉 |, where x1, . . . , xn are unit vectors in H . In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture cn(H) = n n/2 remains open...

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On the Real Linear Polarization Constant Problem

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 linear polarization constant cn(C ) is n, for arbitrary n ∈ N. The same value for cn(R ) is only conjectured. In a recent work A. Pappas and S. Révész prove that cn(R ) = n for n ≤ 5. Moreover, they show that if the linear forms are given as fj(x)...

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The d-th linear polarization constant of R

In [4] it was conjectured that for all unit vectors u1, · · · , ud in Rd, X (u1, · · · , ud) := sup x∈Rd, |x|2=d d ∏ i=1 〈x, ui〉 ≥ 1 with equality occurring iff u1, · · · , ud are orthonormal. We relate this to a conjecture about solutions of Ay = y−1, where A = (〈ui, uj〉), and show that if the conjecture fails then the u1, · · · , ud minimizing X must be linearly dependent. We also show X (u1,...

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ژورنال

عنوان ژورنال: Acta Mathematica Hungarica

سال: 2005

ISSN: 0236-5294,1588-2632

DOI: 10.1007/s10474-005-0214-y