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Is the "Maximum" function any kind of homogeneous or subhomogeneous? Analysis...
Jan 29, 2024
Is the "Maximum" function any kind of homogeneous or subhomogeneous? Analysis Say MM is a map from R∧2R^{\wedge} 2 to R,M(x,y)=max⁡(x,y)R, M(x, y)=\max (x, y), intentionally with no absolute values. Then, are there properties like M(ax,ay)=a∗max⁡(x,y)M(a x, a y)=a^{*} \max (x, y) or <=a^{*} \max (x, y) ? I can't find a reference anywhere either way, I'm hoping the max function has some kind of semi-homogeneous property at least. I'm most interested in what happens with -1 , like if I have M(−x,−y)M(-x,-y). How does this compare to the original max (x,y)(x, y) ?
Solution by Steps
step 1
To determine if the function M(x,y)=max⁡(x,y) M(x, y) = \max(x, y) is homogeneous, we need to check if it satisfies the condition M(ax,ay)=aM(x,y) M(ax, ay) = aM(x, y) for all a∈R a \in \mathbb{R} and for all (x,y)∈R2 (x, y) \in \mathbb{R}^2
step 2
Let's consider the case when x > y . Then M(x,y)=x M(x, y) = x and M(ax,ay)=ax M(ax, ay) = ax . If a a is positive, M(ax,ay)=ax=aM(x,y) M(ax, ay) = ax = aM(x, y) , which satisfies the homogeneity condition
step 3
Now consider the case when y > x . Then M(x,y)=y M(x, y) = y and M(ax,ay)=ay M(ax, ay) = ay . If a a is positive, M(ax,ay)=ay=aM(x,y) M(ax, ay) = ay = aM(x, y) , which also satisfies the homogeneity condition
step 4
However, if a a is negative, the function does not satisfy the homogeneity condition because the max function will switch the argument it takes the maximum over if both x x and y y change sign
step 5
For example, if M(x,y)=x M(x, y) = x because x > y , then for a < 0 , M(ax,ay) M(ax, ay) could be ay ay if |ay| > |ax| , which is not equal to aM(x,y) aM(x, y)
step 6
Therefore, the function M(x,y)=max⁡(x,y) M(x, y) = \max(x, y) is not homogeneous. However, it is positively homogeneous for a > 0 because it satisfies the condition M(ax,ay)=aM(x,y) M(ax, ay) = aM(x, y) in that case
Answer
The function M(x,y)=max⁡(x,y) M(x, y) = \max(x, y) is not homogeneous, but it is positively homogeneous for positive scaling factors.
Key Concept
Homogeneity of a function
Explanation
A function is homogeneous if it satisfies the condition f(ax,ay)=anf(x,y) f(ax, ay) = a^n f(x, y) for all a∈R a \in \mathbb{R} and for all (x,y)∈R2 (x, y) \in \mathbb{R}^2 , where n n is the degree of homogeneity. The max function does not satisfy this condition for negative a a , but it does for positive a a , making it positively homogeneous for positive scaling factors.
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