Concentration on submanifolds of positively curved homogeneous spaces

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.