  Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We
use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant
$\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist
constants $C > c > 0$ independent of $Y$ such that \[c <
\frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here
$\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of
a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then
construct dynamically convex contact forms on $S^3$ that violate this bound
using methods of Abbondandolo-Bramham-Hryniewicz-Salom\~{a}o. These are the
first examples of dynamically convex contact $3$-spheres that are not strictly
contactomorphic to a convex boundary $Y$.

Symplectic Geometry;Differential Geometry;Dynamical Systems