Proving if a function from $\mathbb R^2$ to $ \mathbb R^2$ is convex?

Question

Say $f(x,y)$ = [ $x + y $$\quad$ $ 2x + 7y$ ] , how do I prove that this function is convex?

I know in order to prove that a function from $R^2$ --> $R^1$ is convex then the Hessian of that function has to be pos-semi-def. So is it then sufficient to say that if,

$f(x,y) = x + y$ $\quad$ is convex

and

$f(x,y) = 2x+7y$ $\quad$ is convex,

then

$f(x,y)$ = [$x+y$ $\quad$ $2x+7y$] $\quad$ is also convex?

My intuition says yes, but I haven't been able to find anything supporting this online.

Answer 1

By definition, a function $f: \mathbf{X}\to\bar{\mathbb{R}}$ is convex if $\text{epi}f$ is a convex set. Therefore convexity is a definition which applies to real-valued functions only. Vector-valued functions simply don't posses the property of being be convex or not.