It is not true in general that the function $p$ is concave. For instance, let $\Omega$ be the conical hull of the ball of radius $1$ centered at the point $(2,0,\dots,0)\in\mathbb R^n$. Then $p(x)=c_nx^{n-1}$ for some real $c_n>0$ depending only on $n$ and for all real $x\ge0$. So, $p$ is not concave even on the interval $[0,\infty)$ if $n\ge3$.
However, for $n\ge2$, it is true that the function $p^{1/(n-1)}$ is concave on the interval where it is positive. This follows immediately from the Brunn--Minkowski inequality.