  We establish geometric upper and lower estimates for the Carath\'eodory and
Kobayashi-Eisenman volume elements on the class of non-degenerate convex
domains, as well as on the more general class of non-degenerate
$\mathbb{C}$-convex domains. As a consequence, we obtain explicit universal
lower bounds for the quotient invariant both on non-degenerate convex and
$\mathbb{C}$-convex domains. Here the bounds we derive, for the above mentioned
classes in $\mathbb{C}^{n}$, only depend on the dimension $n$ for a fixed
$n\geq 2$. Finally, it is shown that the Bergman kernel is comparable with
these volume elements up to small/large constants depending only on $n$.
