Show (weak) concavity of a function
I have a real-valued function of variables, defined on the non-negative orthant , which fulfills the following properties:
- The function vanishes on the axes only, i.e., and
- is continuous and twice differentiable
- is marginally concave in all , that is, , and marginally bounded in x_i, that is, is bounded in if all with are held fixed
- For any , one has . Note that one consequence of this is that the concavity can only be weak!
A direct computation of the Hessian does not seem possible. So my question is: can you prove (weak) concavity of this function by knowing only the above properties? (Note that I might have listed more properties than actually needed)
Here is an examplary plot of such a function:
You see very clearly that it is concave, and considering the above properties, it seems like it can't be otherwise.
Thanks for any hints!