On the reverse isoperimetric inequality in Gauss space

Abstract

In this paper, we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in ${\mathbb{R}}^2$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects of the reverse problem have not yet been investigated. In particular, to the best of our knowledge, there seem to be no results on the shape that the isoperimetric set should take. Here, through a local perturbation analysis, we show that smooth perimeter-maximizing sets have locally flat boundaries. Additionally, we derive sharper perimeter bounds than those previously known, particularly for specific classes of convex sets, such as the convex sets symmetric with respect to the axes. Finally, for quadrilaterals with vertices on the coordinate axes, we prove that the set maximizing the perimeter “degenerates” into the $x$-axis, traversed twice.