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Show (weak) concavity of a function
I have a realvalued function of variables, defined on the nonnegative 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:
Attachment 22641
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!
Jens