# Poincar\'e Inequalities and Uniform Rectifiability

## TL;DR

This paper proves that in Euclidean space, Ahlfors regular sets supporting a weak Poincaré inequality are necessarily uniformly rectifiable, linking geometric measure theory with analysis.

## Contribution

It establishes a new characterization of uniform rectifiability via Poincaré inequalities for Ahlfors regular sets in Euclidean space.

## Key findings

- Ahlfors regular sets with Poincaré inequalities are uniformly rectifiable.
- Connects geometric measure theory with analysis through Poincaré inequalities.
- Provides a criterion for uniform rectifiability based on Poincaré inequalities.

## Abstract

We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincar\'e inequality with respect to surface measure is uniformly rectifiable.