Let consider two convex and vector functions of several variables
$f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and $g:\mathbb{R}^n \rightarrow \mathbb{R}^m$.
Somebody can suggest me a reference where I can find the proof that $f + g$ is also convex?
Thank you very much!
Ana
What does it mean that $f\colon \Bbb R^n\to\Bbb R^n$ is convex? There is such notion connected with cones. Let $K\subset\Bbb R^m$ be a cone. Any element $u\in K$ is called nonnegative ($u\ge 0$). We have a natural partial ordering: $u\le v\iff v-u\ge 0\iff v-u\in K$. Then the standard definition of convexity of a real function, which is
$$f\bigl(tx+(1-t)y\bigr)\le tf(x)+(1-t)f(y),$$
or
$$tf(x)+(1-t)f(y)-f\bigl(tx+(1-t)y\bigr)\ge 0,$$
transforms to
$$tf(x)+(1-t)f(y)-f\bigl(tx+(1-t)y\bigr)\in K$$
with standard notation and quantifiers. This is a definition of so-called $K$-convexity of a vector-valued function.
The convexity of $f+g$ is trivial because $K+K\subset K$ (by definition of a cone), which means that $u+v\in K$ whenever $u,v\in K$. No reference is needed, because this is a standard property.