In general, what information is given by $\nabla^2 f \geq 0$?

Question

Since there are many directions one can take when studying this equation, I am curious:

Given a function $f \in C^2$ defined on some open set, what information is given by $\nabla^2 f \geq 0$? Please let me know if I am missing something important for the question to make sense.

Thanks in advance.

Answer 1

It depends on your definition of $\nabla^2 f\ge 0$: If it means that the Hessian matrix of $f$ is positive semidefinite, then $f$ is a convex function.