# Concentration on submanifolds of positively curved homogeneous spaces

## TL;DR

This paper extends Milman's classical concentration result from spheres to positively curved homogeneous spaces, showing Lipschitz functions are nearly constant on high-dimensional submanifolds within these spaces.

## Contribution

It introduces a generalization of measure concentration phenomena from spheres to a broader class of positively curved homogeneous spaces.

## Key findings

- Lipschitz functions are almost constant on high-dimensional submanifolds
- Extension of concentration results beyond spheres
- Applicable to positively curved homogeneous spaces

## Abstract

A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that any Lipschitz function on a positively curved homogeneous space is almost constant on a high dimensional submanifold.