Assume that f is twice differentiable, i.e the Hessian matrix exists at each point in D_f . Then f is convex if and only if D_f is a convex set and its Hessian is positive semi-definite for all x\in D_f . For a single-variable function this reduces to f''(x) \geq 0 . Geometrically this means that f has nonnegative curvature everywhere.
This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.