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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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Once again, this cannot hold in full generality (consider $f\leqslant-1$ $\mu$-almost everywhere). –  Did Jul 24 '12 at 9:46
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Take the 0-Lipschitz function $f=-1$ everywhere: the integral of $f^2$ is $1$, the integral of $f$ is $-1$, hence you are trying to show that $1\leqslant-1+O(1/\sqrt{n})$. // Once your question will be polished, it will be clear that you are after ideas of concentration of measure. In this context, reading (chapter 1 of) Ledoux-Talagrand's book can only be beneficial. –  Did Jul 24 '12 at 10:52
    
@PartialOperator: I've made your latest question into a comment to the user did above. Users with any number of "reputation points" can comment on their own questions and answers (once you obtain 50 points, you gain the ability to comment anywhere), but you were not able to comment because you were not signed into the account that asked the question. I've now merged the duplicate account into the original. If you register your account that should help to prevent future login difficulties. –  Zev Chonoles Jul 25 '12 at 1:20
    
@PartialOperator: Removing the content of your question, as you just did, is considered vandalism here - don't do it again. If you want to delete this question, you can indicate that in a moderator flag (see the "flag" button on the lower left of your question), or (if you register your account) you can delete your own questions and answers. –  Zev Chonoles Jul 25 '12 at 6:11

1 Answer 1

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.

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