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I am looking for references to papers which might have defined a 'signed minimum' equivalent to $$smin(x,y) ::= \left(\frac{\textit{signum}(x)+\textit{signum}(y)}{2}\right)\cdot \min(|x|,|y|) $$ where $\textit(signum)(x)$ is $-1$ for $x\lt 0$, $1$ for $x\gt 0$ and an arbitrary (finite) value for $x=0$. Also, any simpler expression for $smin$ would be appreciated. The above definition works for $\mathbb{R}, \mathbb{Q}$, and $\mathbb{Z}$.

Intuition: the 'signed minimum' between x and y is the one closest to $0$ if they both have the same sign, otherwise it's 0.

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  • $\begingroup$ Why doesn't this work for $\mathbb{Z}$? You can just do case-by case analysis and verify that the result always going to be an integer. $\endgroup$
    – Mikola
    Commented Mar 16, 2010 at 18:13
  • $\begingroup$ It does work, I guess my phrasing is a tad awkward. $\endgroup$
    – Jacques Carette
    Commented Mar 16, 2010 at 19:04

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$smin(x,y)$ can also be described as the number of smallest absolute value in the closed interval between $x$ and $y$.

When $x \le y$ are integers, this is also the value of the game $\{x-1 \mid y+1\}$ in the sense of combinatorial game theory (Conway, On Numbers And Games).

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