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Let $S_1$ and $S_2$ be sets. Let $n_1$ be the cardinality of $S_1$ and $n_2$ be the cardinality of $S_2$. I assume that $n_1$ and $n_2$ are finite. Let $e$ be a function that maps members of $S_1$ and $S_2$ to real numbers. Assume that we have:

$(1/n_1) \sum\limits_{x_i \in S_1} e(x_i) \geq (1/n_2) \sum\limits_{x_i \in S_2} e(x_i)$.

Let $e'$ be a affine transformation of $e$, i.e., we have $e'(x) = ke(x) + l$, where $k$ is positive.

Given this, I know that it is guaranteed that:

$(1/n_1) \sum\limits_{x_i \in S_1} e'(x_i) \geq (1/n_2) \sum\limits_{x_i \in S_2} e'(x_i)$

That is, affine transformations are guaranteed to preserve inequalities between the average values assigned to finite sets by some function $e$.

Question: Is there a class of transformations that isn't a subclass of affine transformations for which this same property is true?

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    I thin$k$ so as well, but can't quite see how to prove it.2012-12-01

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