Hessian criterion for convexity ($g\text{ convex}⟺v^{\mathsf{T}}H(g)(x)v\ge 0$)

On an open convex set, a twice-differentiable real-valued function is convex if and only if its Hessian matrix is a positive semidefinite matrix at every point. In particular continuous second partial derivatives suffice for the differentiability hypothesis. Restrict $g$ to any line segment: $\varphi (t)=g(x+tv)$ has ${\varphi }^{\prime \prime }(t)=v^{\mathsf{T}}H(g)(x+tv)v$. The second-derivative criterion for convexity proves both implications, because convexity is equivalent to convexity of all such restrictions.