A Functional Form of the Isoperimetric Inequality for the Gaussian Measure

Here #n is the standard Gaussian measure in R, of density d#n(x)=>k=1 ,(xk) dxk , x=(x1 , . . ., xn) # R , ,(xk)=1 2? exp(&xk 2), 8 is the inverse of the distribution function 8 of #1 , and A=[x # R: |x&a|<h for some a # A] denotes the open h-neighborhood of A. (1) becomes identity for all half-spaces A of measure p. In these notes we suggest an equivalent analytic form for (1) involving a relation between smooth functions and their derivatives. Relations of such type are well-known for Lebesgue measure (see e.g. [12], Section 3); the Sobolev unequality, for example, provides an equivalent form for the isoperimetric property of balls in the Euclidean space. There is a number of inequalities for the Gaussian measure like Poincare -type or logarithmic Sobolev-type inequalities which can be seen as different versions of socalled ``concentration of (Gaussian) measure phenomenon.'' A question of Article No. 0002