TITLE: An Averaged Version of Gromov's Waist Inequality
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: It is more natural to estimate the total area of large fibres, but it also does not work:https://arxiv.org/abs/1402.2856