Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I hope this forum will be able to help me- 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 $ .

how can i show that such an $f$ satisfies Jensen's Inequality: $( \int_{S^n} f d\mu) ^2 \leq {\int_{S^n} f^2 d\mu } $ ?

in addition, is it true that in such a case we have $ \sqrt{ \int_{S^n} f^2 d\mu } \leq m $ where $m$ is the unique number satisfying $ \mu (f \geq m ) \geq 0.5 , \space \mu (f \leq m ) \geq 0.5 $ ??

hope you'll be able to help !

thanks in advance for any helpful reply

share|improve this question

1 Answer 1

up vote 0 down vote accepted

The first inequality is called Cauchy-Schwarz inequality, rather than Jensen inequality. Its proof is simple and very general: one considers $g=(f-a)^2$ with $a=\int\limits_{S_n}f\,\mathrm d\mu$. Then $g\geqslant0$ everywhere hence $\int\limits_{S_n}g\,\mathrm d\mu\geqslant0$. Expanding this by linearity and using the fact that the mass of $\mu$ is $1$ yields the result.

The second inequality you suggest is odd. If $f(x)=x_1-1$, then $m=-1$, which ruins every chance to get a nonnegative quantity $\leqslant m$. More generally, $ \sqrt{ \int\limits_{S^n} f^2\,\mathrm d\mu } \geqslant \int\limits_{S^n} f\,\mathrm d\mu$ and, as soon as $f$ is symmetric around one of its medians $m$, the RHS is $m$. To sum up, no comparison can exist, and if one existed, it would be the opposite of the one you suggest.

share|improve this answer
    
awsome ! thanks a lot –  Partial Operator Jul 24 '12 at 5:26

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.