# An Averaged Version of Gromov's Waist Inequality

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.

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 is one way around this -- it gives an exact formula for an integral of fiber volumes weighted by the Jacobian of $f$.