5441337177

Yet, differentiating the eąuation whichdefines the function h we get ^^l/n{sL)\_a__t thus h(t) < th'(t). It means that the function h(t)/t is nondecreasing, so 1 = h(l) < h(t)/t for t > 1.    □

For any closed set A the derivative of the function t i—> u_n(tA) is easy to compute. Indeed,

= A [ _e-|*lV2_dJ ₌ f _t2r,_e-t’MV2_dw

t=i J_tA    |_t=1    dt J_A

= 2nv_n{A) — J \z\²du_n(z).

Moreover, the integral of \z\² over a cylinder C may be expressed explicitly in terms of the measure u_n(C). Namely,

/ \z^dv_n{z) = 2(1 — v_n{C)) ln(1 — v_n{C)) + 2nv_n{C).

Je

Combining these two remarks with the preceding proposition we obtain an equivalent formulation of the problem.

Proposition 2. A closed subset K of Cⁿ supports SC-inequality if and only if for any its dilation L

du_n(z) < 2nv_n(L) + 2(1 — v_n{L)) ln (1 — iy_n(L)).

(4)

3 Main result

We aim at proving the aforementioned main result, which reads as follows

Theorem 1. Any set frorn the class 91 supports SC-inequality.

We begin with a one-dimensional entropy ineąuality.

Lemma 1. Let p be a Borel probability measure on R+ and suppose f: R+ —> is a bounded and non-deereasing function. Then

Entnf<—J f{x) ^1 + \np ((x, oo)) ^d/Lf(x).    (5)

Proof. Using homogeneity of both sides of (5), without loss of generality, we can assume that /_R+ fdp = 1. Then we may rewrite the assertion of the lemma as follows

/ ln ( /(^) / dp(t) ]f(x)dp(x) < -1.

J k+    V ■'(*.«>)    /

Introduce the probability measure u on M+ with the density / with respect to p. Thanks to the monotonicity of / we can bound the left hand side of the last ineąuality by

ln(v((x,oo))'jdv(x) = -^l{_u>„((_x,oo ))>(«, ®)di/(®).

3