Consider the following Hamiltonian:
$H(x,p,c)=U(c)+p((a+b-c)x-c)$
Asume $U(c)$ is concave in $c$, then the hamiltonian is also concave in $c$. Here $x$ denotes the state and $p$ the adjoint variable.
But i want to show that the maximized Hamiltonian defined by
$H^\circ(x,p)=\max_{c \in C}H(x,p,c)$
is concave in the state $x$.
I know this is true if $H$ is concave in $c$ and $x$, but it think this is not the case here. Maybe it is, but i can't see it.
Any other ways to show that $H^\circ$ is concave in $x$?
I know this is definitely the case, but i am unable to proof this.