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

Studia Mathematica (1993)

• Volume: 105, Issue: 2, page 173-187
• ISSN: 0039-3223

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## Abstract

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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.

## How to cite

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Kwapień, Stanisław, and Sawa, Jerzy. "On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets." Studia Mathematica 105.2 (1993): 173-187. <http://eudml.org/doc/215993>.

@article{Kwapień1993,
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.},
author = {Kwapień, Stanisław, Sawa, Jerzy},
journal = {Studia Mathematica},
keywords = {dilatations of convex symmetric sets; Gaussian measure on a Banach space; Gaussian measures of large and small balls; Hilbert space},
language = {eng},
number = {2},
pages = {173-187},
title = {On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets},
url = {http://eudml.org/doc/215993},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Kwapień, Stanisław
AU - Sawa, Jerzy
TI - On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 2
SP - 173
EP - 187
AB - 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.
LA - eng
KW - dilatations of convex symmetric sets; Gaussian measure on a Banach space; Gaussian measures of large and small balls; Hilbert space
UR - http://eudml.org/doc/215993
ER -

## References

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1. [1] N. K. Bakirov, Extremal distributions of quadratic forms of gaussian variables, Teor. Veroyatnost. i Primenen. 34 (1989), 241-250 (in Russian). Zbl0675.60016
2. [2] T. Byczkowski, Remarks on Gaussian isoperimetry, preprint, Wrocław University of Technology, 1991.
3. [3] A. Ehrhard, Symétrisation dans l'espace de Gauss, Math. Scand. 53 (1983), 281-381.
4. [4] H. J. Landau and L. A. Shepp, On the supremum of a Gaussian process, Sankhyā Ser. A 32 (1970), 369-378. Zbl0218.60039
5. [5] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, 1991.
6. [6] S. Kwapień and J. Sawa, On minimal volume of the convex hull of a set with fixed area on the sphere, preprint, Warsaw University, to appear. Zbl0810.60035
7. [7] S. J. Szarek, Condition numbers of random matrices, J. Complexity 7 (1991), 131-149. Zbl0760.15018
8. [8] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, Reidel, Dordrecht 1987.

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