Suppose $x,y \in \mathbb{R_+}, x<y$, and $ 0 < \varepsilon \leq (y-x)/2$. It seems to me that $xy < (x+\varepsilon)(y-\varepsilon)$ and equivalently that $(x+\varepsilon)(y-\varepsilon)$ is strictly increasing in epsilon for $ 0 \leq \varepsilon \leq (y-x)/2$.

Is there a name for this property? More generally, is there a name for this kind of property of a function where moving the inputs closer together while preserving their sum (or perhaps preserving some other analogous property) increases the function?

**Edited from previous error:**As example of a similar (but opposite) property, it looks like the euclidean norm of a two dimensional vector (this probably could be extended to higher dimensions) is decreasing as the coordinates are brought closer together while keeping their sum (i.e. the taxicab norm) constant.

It seems like this has something to do with convexity but I'm not sure what the relationship is and if it is directly related or more of an analogy.

  • 1
    $\begingroup$ Your third paragraph is false. I doubt that there is any name for this. $\endgroup$
    – user24142
    Aug 31, 2015 at 22:53
  • $\begingroup$ whoops yeah said it the wrong way around in the 3rd paragraph I'll edit $\endgroup$
    – Smithey
    Sep 1, 2015 at 0:22


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