On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets

Stanisław Kwapień; Jerzy Sawa

ArticleOriginal scientific text

Title

On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets

Authors ¹, ¹

Affiliations

Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Abstract

The paper deals with the following conjecture: if μ is a centered Gaussian measure on a Banach space F,λ > 1, K ⊂ F is a convex, symmetric, closed set, P ⊂ F is a symmetric strip, i.e. P = {x ∈ F : |x'(x)| ≤ 1} for some x' ∈ F', such that μ(K) = μ(P) then μ(λK) ≥ μ(λP). We prove that the conjecture is true under the additional assumption that K is "sufficiently symmetric" with respect to μ, in particular it is true when K is a ball in Hilbert space. As an application we give estimates of Gaussian measures of large and small balls in a Hilbert space.

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Title

On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets

Affiliations

Abstract

Bibliography