It is not true in general that the function $p$ is concave. 

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][1]. 


  [1]: https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski_theorem#Remarks