TITLE: what's the property of concat function?
QUESTION [0 upvotes]: For concatenation functions such as $f(\vec{p}) =(\vec{p},\vec{p})$ or $g(\vec{p}) =(\vec{p},\vec{1}-\vec{p})$. where $\vec{p}, \vec{1}-\vec{p} \in P \subseteq [0,1]^n $.
What's the property of it? is it convex? how should I approach it? 
What my context is that $\vec{p} \in P$ where $P$ is a convex set.
And I have a convex function $l$ where ${\displaystyle l\colon \,P\times P\mapsto \mathbb {R}}$.
For the composed function $l(g(\vec{p}))$ can i claim that  
$
{\displaystyle {\underset {\vec{x}\in P}{\operatorname {arg\,max} }}\,l(g(\vec{x}))\in \text{closure of} convexhull(P) }
$ ?

REPLY [1 votes]: For concatenation functions such as $f(\vec{p}) =(\vec{p},\vec{p})$ or $g(\vec{p}) =(\vec{p},\vec{1}-\vec{p})$. where $\vec{p}, \vec{1}-\vec{p} \in P \subseteq [0,1]^n $.
  What's the property of it? is it convex?

Vector-valued functions do not have a notion of convexity, so no.

For the composed function $l(g(\vec{p}))$ can i claim that
$
{\displaystyle {\underset {\vec{x}\in P}{\operatorname {arg\,max} }}\,l(g(\vec{x}))\in convexhull(P) }
$ ?

By definition, if you take the argmax over $\vec{x} \in P$, you will get out an element $\vec{x}_0 \in P$. Since $P$ is convex, it is equal to its convex hull and the result trivially holds.
Perhaps you want to know whether $l(f(\vec{p}))$ or $l(g(\vec{p}))$ are convex. That is, does it hold that
$$l(f(t\vec{p}_1+(1-t)\vec{p}_2)) \leq t l(f(\vec{p}_1)) + (1-t) l(f(\vec{p}_2)) \forall t \in [0,1]$$
and similarly for $g$. For $f$, notice that
$$f(t\vec{p}_1+(1-t)\vec{p}_2) = (t\vec{p}_1+(1-t)\vec{p}_2,t\vec{p}_1+(1-t)\vec{p}_2)= t(\vec{p}_1,\vec{p}_1)+(1-t)(\vec{p}_2,\vec{p}_2).$$
Now use the convexity of $l$.
For $g$ it is a little trickier, but you can show that
\begin{align*}
g(t\vec{p}_1,(1-t)\vec{p}_2)&=(t\vec{p}_1+(1-t)\vec{p}_2, \vec{1}-(t\vec{p}_1+(1-t)\vec{p}_2))\\
&= (t\vec{p}_1+(1-t)\vec{p}_2, t (\vec{1}-\vec{p}_1) + (1-t)(\vec{1}-\vec{p}_2))\\
&= t(\vec{p}_1, \vec{1}-\vec{p}_1) + (1-t) (\vec{p}_2, \vec{1}-\vec{p}_2).
\end{align*}
Now again use the convexity of $l$.
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