On the reverse isoperimetric inequality in Gauss space

TL;DR

The paper tackles the reverse Gaussian isoperimetric problem for convex sets in $\mathbb{R}^2$, aiming to identify shapes that maximize Gaussian perimeter under convexity constraints. It employs local perturbation and variational analyses to show that maximizers must have locally flat boundaries and derives sharp bounds within special convex classes. Specifically, it proves $P_{\gamma_2}(T)<\sqrt{2/\pi}$ for axis-vertex quadrilaterals in $\widetilde{T}$, with rhombi degenerating to the $x$-axis achieving the bound asymptotically, and establishes a universal bound $P_{\gamma_2}(C)\le 2/\sqrt{\pi}$ for the broader class $\widetilde{C}$. The results refine constants in Gaussian-space shape optimization and illuminate the likely geometry of extremal sets, with potential connections to learning theory and related applications.

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.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Proposition 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4