Relation between $p$-superharmonic functions and concave functions

Question

If I understood correctly what I read. In the one-dimensional situation the $p$ -superharmonic functions are exactly the concave functions and in several dimensions, the concave functions are $p$-superharmonic, simultaneously for all $p$, but there are $p$-superharmonic functions that it aren't concave. How can I see this?

Answer 1

Ley $f(x,y)=x^2-y^2$. Then $f$ is a $2$-harmonic function. It is not concave, nor convex.

Remark: Every $p$-harmonic function is $p$-subharmonic and $p$-superharmonic.