ON SOME CONJECTURE CONCERNING GAUSSIAN MEASURES OF DILATATIONS OF CONVEX SYMMETRICAL SETS

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.