# Expectation and Median (Jensen's inequality) of Spacial Functions

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 !

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.