On the Gaussian Concentration Inequality and Its Relation to the Gaussian Surface Area (preliminary Version)

Let γ2 be a standard Gaussian measure in R. For a given measurable set Q in R, let HQ be a half space in R such that γ2(Q) = γ2(HQ). The classical Gaussian concentration inequality states that for all measurable sets Q ⊂ R and for all h > 0, γ2(Q+ hB n 2 ) ≥ γ2(HQ + hB 2 ). Under some minor constrains on the set Q we obtain an improvement of the latter inequality in a certain range of h, depending on the Gaussian surface area of Q.