Inserted: 15 may 2017
Concentration of measure is a principle that informally states that in some spaces any Lipschitz function is essentially constant on a set of almost full measure. From a geometric point of view, it is very important to find some structured subsets on which this phenomenon occurs. In this paper, I generalize a well-known result on the sphere due to Milman to a class of Riemannian manifolds. I prove that any Lipschitz function on a compact, positively curved, homogeneous space is almost constant on a high dimensional submanifold.