In convex analysis, the second-order condition for strictly convex functions has a distinct loss of symmetry in that the implication is only one way.
Recall that: a twice differentiable function $f$, $\nabla^2 f(x) \succ 0$ (positive definite) for all $x$ in the domain of $f$ implies that $f$ is strictly convex.
This seems to be a "one-off" in the sense that the second-order condition for convexity and strong convexity are all if and only if. Similarly, the first-order condition for convexity, strict and strong convexity are all if and only if. Why the sudden loss of symmetry?
Is there a intuitive way of thinking about (or explaining) why possessing positive definite Hessian is not sufficient for strict convexity?