I want to continue a bit my earlier post: Expectation and median (Jensen’s inequality) of spacial functions
So, if we have a 1-Lipschitz function $f:S^n \to \mathbb{R} $ , when $S^n$ is equipped with the geodesic distance d and with the uniform measure $ \mu $ , it's pretty easy to show that the median of such a function can be estimated by: $ |m- \int_{S^n} f d\mu | =O(\frac{1}{\sqrt{n}} ) $ . How can one show that we also have $ \sqrt{ \int_{S^n} f^2 d\mu } \leq \int_{S^n} f d\mu+ O(\frac{1}{\sqrt{n}}) $ where the constant under both of the big-O's is the same one?
does someone have an idea?
thanks !