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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 !

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The proposed inequality $\sqrt{ \int_{S^n} f^2 d\mu } \leq \int_{S^n} f d\mu+ O\left(\frac{1}{\sqrt{n}}\right) \tag1$ was disproved by Did in a comment: take $f\equiv -1$ (which is $0$-Lipschitz), then (1) becomes $1\le -1+O(1/\sqrt{n})$, which is false. Quoting more of the comment:

... you are after ideas of concentration of measure. In this context, reading (chapter 1 of) Ledoux-Talagrand's book can only be beneficial.