LINEAR POLARIZATION CONSTANT OF ℝ^{n}

MÁTÉ MATOLCSI

In this note we aim to make a contribution to estimating the nth linear polarization constant c_{n}(H) of an n-dimensional real Hilbert space H. We recall the

Definition 0.1. (Benítez, Sarantopoulos, Tonge) The n-th linear polarization constant of a Banach space X is defined by

c_{n}(X) := inf{M : ∥f_{1}∥ ⋯ ∥f_{n}∥ ≤ M∥f_{1} ⋯ f_{n}∥ (∀f_{1}, …, f_{n} ∈ X^{∗})}
= 1/ inf_{f_{1},...,f_{n}∈S_{X^{∗}}} sup_{∥x∥=1} |f_{1}(x) ⋯ f_{n}(x)|.

For a real or complex Hilbert space H of dimension at least n, we always have c_{n}(H) = c_{n}(𝕂^{n}).

Benítez, Sarantopoulos and Tonge proved that for isomorphic Banach spaces X and Y we have c_{n}(X) ≤ d^{n}(X, Y)c_{n}(Y), where d(X, Y) denotes the Banach-Mazur distance of X and Y. Note, that for any n-dimensional space X a result of John states that d(X, 𝕂^{n}) ≤ \sqrt{n} (where 𝕂^{n} denotes the n-dimensional Hilbert space). The combination of these results mean that the determination of c_{n}(𝕂^{n}) gives information on the linear polarization constants of other spaces, too.

The exact values of c_{m}(𝕂^{n}) seem very hard to determine. A remarkable result of Arias-de-Reyna states that c_{n}(ℂ^{n}) = n^{n/2}. Ball’s recent solution of the complex plank problem also implies the same result.

The value of c_{n}(ℝ^{n}) seems even harder to find. The determination of c_{n}(ℝ^{n}), by the definition and the Riesz representation theorem, boils down to determining

I := inf_{x_{1},...,x_{n}∈S} sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ |

The estimate I ≤ n^{− \frac{n}{2}} follows by considering an orthonormal system.

The result of Arias-de-Reyna can be used to derive the following estimates:

n^{\frac{n}{2}} ≤ c_{n}(ℝ^{n}) ≤ 2^{\frac{n}{2} −1}n^{\frac{n}{2}} .

A natural, intriguing conjecture is the following.

Conjecture. c_{n}(ℝ^{n}) = n^{n/2}.

Marcus gives the following estimate: If x_{1}, x_{2}, …, x_{n} are unit vectors in ℝ^{n} then there exists a unit vector y such that

(1)  | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | ≥ (λ_{1}/n)^{n/2},

where λ_{1} denotes the smallest eigenvalue of the Gram matrix XX^{∗} = [⟨x_{i}, x_{j}⟩]. Marcus also expressed the opinion that lower bounds on sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | should involve the eigenvalues λ_{1}, …, λ_{n} of the Gram matrix XX^{∗}, i.e. we should look for estimates of the form sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | ≥ f(λ_{1}, …, λ_{n})n^{−n/2}. Note that ∑_{j} λ_{j} = Tr XX^{∗} = n. Therefore the above Conjecture can be equivalently formulated as

sup_{∥y∥=1} | ⟨x_{1}, y⟩ ⋯ ⟨x_{n}, y⟩ | ≥ 1 · n^{−n/2} = (\frac{λ_{1} + ⋯ + λ_{n}}{n} )^{n/2} n^{−n/2}.

In this note we show that if we replace the arithmetic mean by the harmonic mean of the numbers λ_{1}, …, λ_{n}, then the corresponding lower estimate does hold.

Theorem 0.1. Assume that the given unit vectors x_{1}, …, x_{n} are linearly independent. Let λ_{1}, …, λ_{n} denote the eigenvalues of the Gram matrix XX^{∗}. Then

(2)  max_{∥y∥=1} ∏_{j=1}^{n} |(Xy)_{j}| ≥ (\frac{n}{λ_{1}^{−1} + ⋯ + λ_{n}^{−1}} )^{n/2} · n^{n/2}

This gives an improvement of Marcus’ result but the Conjecture still remains open.