Gromov's waist inequality for unit n-sphere $\mathbb{S}^{n}$ says:
For any continuous function $f: \mathbb{S}^{n} \rightarrow \mathbb{R}^{m} $, 
there is some $y \in \mathbb{R}^{m}$ s.t. $Vol\_{n-m}(f^{-1}(y)) \geq Vol\_{n-m}(\mathbb{S}^{n-m}) $. 


I'm wondering if there is an averaged version of the inequality, 
comparing the averaged fiber volume and some averaged $\mathbb{S}^{n-m}$ volume. 
For example, 
is it true that for some constant $C(n, m)$ depending only on dimensions:


$ \int\_{f(\mathbb{S}^{n})} Vol\_{n-m}(f^{-1}(y)) 
\geq C(n, m) Vol\_{m}(f(\mathbb{S}^{n})) Vol\_{n-m}(\mathbb{S}^{n-m}) $ 


It is, of course, interesting to consider the tubular version:


$ \int\_{f(\mathbb{S}^{n})} Vol\_{n}(f^{-1}(y) + \epsilon ) 
\geq C(n, m) Vol\_{m}(f(\mathbb{S}^{n})) Vol\_{n}(\mathbb{S}^{n-m} + \epsilon ) $, as well as similar inequalities for ball, cube, etc. in place of sphere.



An Averaged Version of Gromov's Waist Inequality
My guess would be that there are ways to manipulate the average -- say you take the unit sphere in $\mathbb{R}^3$, and a function like $f(x,y,z)=x^{100}$. Then the image is the interval $[-1,1]$, but the preimage of a random point is small. The [coarea formula](https://en.wikipedia.org/wiki/Coarea_formula) is one way around this -- it gives an exact formula for an integral of fiber volumes weighted by the Jacobian of $f$.


