S. Kwapien et J. Sawa, ON SOME CONJECTURE CONCERNING GAUSSIAN MEASURES OF DILATATIONS OF CONVEX SYMMETRICAL SETS, Studia Mathematica, 105(2), 1993, pp. 173-187
The paper deals with the following conjecture: if mu is a centered Gau
ssian measure on a Banach space F, lambda > 1, K subset-of F is a conv
ex, symmetric, closed set, P subset-of F is a symmetric strip, i.e. P
= {x is-an-element-of F : \x'(x)\ less-than-or-equal-to 1} for some x'
is-an-element-of F', such that mu(K) = mu(P) then mu(lambdaK) greater
-than-or-equal-to mu(lambdaP). We prove that the conjecture is true un
der the additional assumption that K is ''sufficiently symmetric'' wit
h respect to mu, in particular it is true when K is a ball in a Hilber
t space. As an application we give estimates of Gaussian measures of l
arge and small balls in a Hilbert space.