Convex function's monotonicity?

Question

Let $F:[0, \infty)^n\rightarrow [0, \infty)$ be a convex function and $F(x_1,x_2,...,x_n)=0$ if and only if $x_i=0$ for $i=1,...,n$.

Is $F$ a nondecreasing function, i.e if $x_i\leq y_i$ for $i=1,...,n$, then $F(x_1,x_2,...,x_n)\leq F(y_1,y_2,...,y_n)$?

If not do you have any counterexample in mind?

Thank you in advance

Answer 1

Not necessarily, for example take $F(x) = (x_1-x_2)^2+0.1x_2$, $x=(1,0)$, $y=(2,2)$.