Gaussian-type Isoperimetric Inequalities in $RCD(K,\infty)$ probability spaces for positive $K$

TL;DR

This paper extends isoperimetric inequalities to $RCD(K, abla)$ spaces using $ ext{Gamma}$-calculus, establishing Gaussian inequalities for positive $K$, thus broadening geometric analysis in metric measure spaces.

Contribution

It adapts $ ext{Gamma}$-calculus techniques to $RCD(K, abla)$ spaces and proves Gaussian isoperimetric inequalities for positive $K$, a novel extension in this setting.

Findings

Proved Bobkov's local isoperimetric inequality in $RCD(K, abla)$ spaces.

Established Gaussian isoperimetric inequality for positive $K$ in these spaces.

Utilized measure-valued $ ext{Gamma}_2$ operator for the proofs.

Abstract

In this paper we adapt the well-estabilished $\Gamma$-calculus techniques to the context of $RCD(K,\infty)$ spaces, proving Bobkov's local isoperimetric inequality and, when $K$ is positive, the Gaussian isoperimetric inequality in this class of spaces. The proof relies on the measure-valued $\Gamma_2$ operator introduced by Savar\'e.