convex, twice-differentiable vector-values function properties

Question

For any convex, twice-differentiable vector-values function f:R^n -> R^m, it's n*n Hessian Matrix is positive semi-definate

Is the above statement True or False? Explain

Answer 1

How to define a convex function $f:\mathbb{R}^n\to\mathbb{R}^m$ for $m>1$? Some inequality is needed. If $f:\Bbb{R}^n\to\Bbb{R}$, that the Hessian matrix is positive semi-definite is the equivalent condition of convexity. How to explain this? See the proof.