The reverse isoperimetric problem for Gaussian measure (Q1312191)

It is shown that if \(g\) is the standard Gaussian density on \(\mathbb{R}^ n\) and \(C\) is a convex body in \(\mathbb{R}^ n\), then \[ \int_{\partial C}g \leq 4n^{1/4} \] (the integral is taken w.r.t the \((n-1)\)-dimensional Hausdorff measure on the boundary \(\partial C)\). The method used is suggested by Cauchy's formula for the surface area of convex bodies. The corresponding estimate from below is \(\sqrt{\log n}/e\) (for \(C\) being a centrally symmetric cube). The arguments presented raise several questions in integral geometry.