POLARIZATION CONSTANTS, PLANK PROBLEMS, AND LOCAL THEORY OF BANACH SPACES

SZILÁRD GY. RÉVÉSZ^{∗} AND YANNIS SARANTOPOULOS

Definition 1 (Benítez - Sarantopulos - Tonge). The n^{th} (linear) polarization constant of a Banach space X is defined by

c_{n}(X) := inf{M > 0 : ∥f_{1}∥ ⋯ ∥f_{n}∥ ≤ M · ∥f_{1} ⋯ f_{n}∥, ∀f_{1}, …, f_{n} ∈ X^{∗}} . (1)

Moreover, the (linear) polarization constant of X is c(X) := lim sup_{n→∞} c_{n}(X)^{1/n}.

If f_{1}, f_{2}, …, f_{n} ∈ X^{∗}, then the product (f_{1}f_{2} ⋯ f_{n})(x) := f_{1}(x)f_{2}(x) ⋯ f_{n}(x) is a continuous n-homogeneous polynomial on X. In 1998 R.A. Ryan and B. Turett proved the existence of a universal bound C_{n} of (1), and Benítez, Sarantopoulos and Tonge proved that in the case of complex Banach spaces, C_{n} ≤ n^{n} and the constant n^{n} is best possible. Clearly, c_{n}(·) and c(·) are isometric properties of Banach spaces, also fitting into the local theory of Banach spaces, since for any X we have c_{n}(X) = sup {c_{n}(Y) : Y ⊆ X, dim Y = n}.

Proposition 1. For c(X) also the limit exists and we have c(X) = lim_{n→∞} c_{n}(X)^{1/n} .

The growth of c_{n}(X) when n → ∞ is characteristic to the space X. Eg. we have

Theorem 2. Let X be any normed space. Then c(X) = ∞ iff dim(X) = ∞.

Although it seems to be very likely, but it is not known whether c_{n}(X) ≥ c_{n}(ℓ_{2}^{n}) (∀n ≤ dim X) holds true in general. Still, for each n ∈ IN, Hilbert spaces have the smallest n^{th} polarization constant among infinite dimensional normed spaces.

Theorem 3. If dim X = ∞, then c_{n}(X) ≥ c_{n}(ℓ_{2}^{n}) , ∀n ∈ IN.

In this work we discuss various results, estimates and exact values found for linear polarization constants of Banach, and in particular of Hilbert spaces. We also present expected and unexpected connections to the plank problem of Tarski, to Chebyshev constants, and to various well-known polynomial inequalities. In this respect, one of the challanging problems of the subject is the following conjecture, which seems to be widely believed.

Conjecture [Sarantopoulos]. c_{n}(ℝ^{n}) = n^{\frac{n}{2}} .

Note that Arias-de Reyna proved c_{n}(ℂ^{n}) = n^{\frac{n}{2}} , and an even more precise result was found by K. Ball recently. We will explain, why the real case is more difficult.

(Szilárd Gy. Révész) A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary.

(Yannis Sarantopoulos) National Technical University, School of Applied Mathematical and Physical Sciences, Department of Mathematics, Zografou Campus 157 80, Athens, Greece.