# Rigidity of the 1-Bakry-\'Emery inequality and sets of finite perimeter in RCD spaces

## TL;DR

This paper investigates the asymptotic geometric structure of sets with finite perimeter in RCD spaces, establishing tangent half-spaces almost everywhere and characterizing special RCD spaces via the 1-Bakry-Émery inequality.

## Contribution

It proves the existence of tangent half-spaces almost everywhere in RCD spaces and characterizes RCD(0,N) spaces with equality in the 1-Bakry-Émery inequality, providing new insights even in smooth cases.

## Key findings

- Existence of Euclidean tangent half-spaces almost everywhere.
- Uniqueness of tangent half-spaces in non-collapsed RCD spaces.
- Complete characterization of RCD(0,N) spaces with equality in the 1-Bakry-Émery inequality.

## Abstract

This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over RCD(K,N) metric measure spaces. Our main result asserts existence of a Euclidean tangent half-space almost everywhere with respect to the perimeter measure and it can be improved to an existence and uniqueness statement when the ambient is non collapsed. As an intermediate tool, we provide a complete characterization of the class of RCD(0,N) spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry-\'Emery inequality. This result is of independent interest and it is new, up to our knowledge, even in the smooth framework.