I am currently working on weak optimal controls and I got stuck on the so called roxin's condtion. The condition stats in my case that the set
$$M:=\lbrace (f(u), |u|^2 |u\in U\rbrace,$$
where f(u) is linear in u, |.| is the euklidian norm of u and U is a convex compact set, is also convex. However, I struggel to show that the set $M$ is convex. For f it is easy because of the linearity it is an affine mapping however for the quadratic part I am unsure. This is how far I got: We have to show that for $\lambda \in [0, 1]$ and $m,m'\in M$, $$\lambda m +(1-\lambda)m' \in M$$ \begin{align} \lambda \begin{pmatrix}f(u)\\ |u|^2 \end{pmatrix} +(1-\lambda)\begin{pmatrix}f(u')\\ |u'|^2 \end{pmatrix} =\begin{pmatrix} f(\lambda u+(1-\lambda)u')\\ \lambda u^2+(1-\lambda)|u'|^2 \end{pmatrix}, \end{align} where $\lambda u+(1-\lambda)u'\in U$ because U is convex. However, I can not get it to work for the square of the norm. Maybe somebody has an idea.
Big thanks in advance!
In general your set, which is like $\{(x,x^2),\ x\in[0,1]\}$, will not be convex; however if your problem is comming from an optimisation problem, I think it is sometimes only needed that an other set be convex (assuming that you can recast you original problem into an equivalent one with adding a control $\eta\geq 0$) that is, the set $\{(x,x^2+\eta),\ x\in[0,1], \eta\geq 0\}$ be convex or possibly $\{(x, (x^2+\eta)\wedge 1),\ x\in[0,1], \eta\geq 0\}$ be convex (and these sets are indeed convex).