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

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A suitable concave function in two variables, X and Y, for modeling is proposed as f(x,y) = ln(x + 1/2) + ln(y + 1/2). This function meets the specified conditions: it is negative for x<0.5 and/or y<0.5, equals zero at x=0.5 and y=0.5, and is positive for x>0.5 and y>0.5. Additionally, it maintains concavity since the second order partial derivatives with respect to X and Y are both less than zero. The discussion highlights the need for a mathematical function that adheres to these constraints for effective modeling. The proposed function appears to satisfy all requirements outlined in the query.
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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
 
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Hi johnd17! Welcome to PF! :smile:

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

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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