What is a Suitable Concave Function in 2 Variables for Modelling?

[johnd17]

I need a concave function in 2 variables X and Y for some modelling I'm doing.

For 0<x<1 and 0<y<1, the function needs to satisfy the following:

f(x,y) < 0 for x<0.5 and/or y<0.5
f(x,y) = 0 for x=0.5 and y=0.5
f(x,y) > 0 for x>0.5 and y>0.5

MUST be concave.. i.e. second order partials wrt X and Y must both be < 0.

My maths was never that good and I can't seem to make any progress here.. can anyone help out? Thanks

 

[I like Serena]

Hi johnd17! Welcome to PF!

What about:
$$f(x,y) = \ln(x + \frac 1 2) + \ln(y + \frac 1 2)$$