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.