  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.
