# Volume Doubling Condition and a Local Poincar\'e Inequality on Unweighted Random Geometric Graphs

## TL;DR

This paper proves that certain random geometric graphs built from points on a submanifold satisfy key geometric and functional inequalities, specifically volume doubling and local Poincaré inequalities, with high probability.

## Contribution

It establishes the conditions under which volume doubling and local Poincaré inequalities hold for unweighted random geometric graphs on submanifolds.

## Key findings

- Volume doubling condition holds with high probability.
- Local Poincaré inequality is valid under regularity conditions.
- Results apply uniformly over all shortest path balls within a certain radius.

## Abstract

The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider $\varepsilon$-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of $\mathbb{R}^K$. We show that a volume doubling condition (VD) and local Poincar\'e inequality (LPI) hold for the random geometric graph (with high probability, and uniformly over all shortest path distance balls in a certain radius range) under suitable regularity conditions of the underlying submanifold and the sampling distribution.