For a closed smooth oriented manifold $M$, define $$ \text{Minvol}(M) = \inf_{g: \substack{|K_g| \le 1\\g \text{ complete}}} \text{vol}_g(M), $$ where $|K_g|$ here denotes $\sup_{p \in M}\sup_{\Pi \text{ plane in }T_pM}|K_g(\Pi)|$ (whre $K_g$ is the sectional curvature).
In this paper by Gromov (top of Pg 3 of the PDF aka Pg 6 of the File), it is stated that "by Gauss-Bonnet", for every $n$, for we have some constants $c_n > 0$, $c'_n > 0$ such that for every closed smooth (oriented) $M$ of dimension $n$,
- $\text{Minvol}(M) \ge c_n|\chi(M)|$,
- $\text{Minvol}(M) \ge c'_n|p(M)|$ for the Pontyagin numbers $p(M)$ of $M$.
The question is: Apart from the most naive case (i.e. $n = 2$, which applies to case 1.), how do we see this? Is the proof written down anywhere?