5441337176

2 Preliminaries

We define the standard Gaussian measure u_n on the space Cⁿ via the formula

v_n(A) = 72_n (t(A)) , for any Borel set AcCⁿ,

where Cⁿ >-E> R²ⁿ is the bijection given by

t(zi, ..., z_n) = (9\zzi,3mzi,..., 91ez_n,2Jm2_n).

We adopt the notation M₊ = [0, +oo). Later on we will also extensively use the notion of the entropy of a function /: X —> M+ with respect to a probability measure p on a measurable space X

Ent_M/= j^/(x)ln/(s)d/i(a)-    f(x)d/i(x)^ ln    f(x)d/i(x) V (1)

We say that a closed subset K of Cⁿ supports the complex S-inequality, SC-inequality for short, if any its dilation L = sK, s > 0, and any cylinder C = {z € Cⁿ | \zi\ < R} satisfy

u_n(L) = u_n(C) ==> i'_n{tL) > v_n(tC), for t>l.    (2)

Notę that the natural counterpart of 5-inequality in the complex case is the fol-lowing conjecture due to Prof. A. Pełczyński, which has already been discussed in [Tko].

Conjecture. Ali closed subsets K of Cⁿ which are rotationally symmetric, that is e^l9K = K for any 9 & R, support SC-inequality.

In the present paper we are interested in the class of all closed sets in Cⁿ which are Reinhardt complete, i.e. along with each point (z\,... ,z_n) such a set contains all points (w\,..., w_n) for which liufcl < |zfc|, k = 1,,n (consult for instance the textbook [Sh, 1.1.2, pp. 8-9]). The key point is that this class contains all unit balls with respect to unconditional norms on Cⁿ. Recall that a norm || • || is said to be unconditional if \\(e^t9lzi,... ,e^ienz_n)\\ = ||2|| for all 2 G Cⁿ and 9\,..., 9_n € M.

The goal is to prove that all sets from the class 91 support SC-inequality. Now we establish some generał yet simple observations which allow us to reduce the problem to a one-dimensional entropy inequality.

Proposition 1. A closed subset K of Cⁿ supports SC-inequality if and only if for any its dilation L and any cylinder C we have

"n(£) = V_n{G) =>•    ^n(tL)\ >f.Vn(tC) I .    (3)

^    \t=l    lt=l

Proof. We are only to show the interesting part that (3) implies (2) following the proof of [KS, Lemma 1]. Fix a dilation L of K and a cylinder C such that u_n(L) = u_n(C). Let a function h be given by u_n(tL) = v_n(h(t)C), t > 1. Then, by the assumption, we find

h(t)-^-v_n(sC)\    =

^ds    I s=h(t) ^ds    \s=i    ^ds    l«=i ^ds    \s=t

2