International Congress of Mathematicians

814 R. Latalathat a(x*) = 0 for ail x* £ F*. A random vector with values in F is said to beGaussian if its distribution is Gaussian. Every centered Gaussian measure on R"is a linear image of the canonical Gaussian measure -y n , that is the measure onR" with the density dj n (x) = (27r) - "/ 2 exp(^|x| 2 /2)dx, where |x| = \/Y^i=ix ì-Infinite dimensional Gaussian measures can be effectively approximated by finitedimensional ones using the following series representation (cf. [18, Proposition 4.2]):If p is a centered Gaussian measure on F and gi,g 2 , • • • are independent A'(0,1)random variables then there exist vectors xi, x 2 ,... in F such that the series X =Si-i x ì9ì ' 1S convergent almost surely and in every IP, 0 < p < oo, and is distributedas p.We will denote by $ the distribution function of the standard normal A'(0,1)r.v., that is1 f' x$(x) = 7i(—oo,x) = / er y ' 2 dy, ^oo < x < oo.V 2-ÏÏ i-ooFor two sets A,B in a Banach space F and t £ M. we will write tA = {tx : x £ A}and A + B = {x + y : x £ A,y £ B}. A set A in F is said to be symmetric if-A = A.Alany results presented in this note can be generalized to the more generalcase of Radon Gaussian measures on locally convex spaces. For precise definitionssee [4] or [7].2. Gaussian isoperimetryFor a Borei set A in R" and t > 0 let A t = A + tB% = {x £ R" : |x - a\ 0. (2.1)Theorem 2.1 has an equivalent differential analog. To state it let us definefor a measure p on R" and any Borei set A the boundary p-measure of A by theformulaM+(A)=liminf M( - 4t)^(- 4) .Aloreover let ip(x) = $'(x) = (27T) -1 / 2 exp(^x 2 /2) and letI(t) = V>°$- 1 (t),t£[0,l]be the Gaussian isoperimetricfunction.