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As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function.

$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$

There's even a nice intuitive explanation to go along with this: If we go to the point half way between two numbers, then going down by half their difference will take us to the smaller one. So my question is: "Is there a similar formula for three numbers?"

Obviously $\min(a,\min(b,c))$ will work, but this gives us the expression: $$\frac{a+\left(\frac{b+c}{2} - \frac{|b-c|}{2}\right)}{2} - \frac{\left|a-\left(\frac{b+c}{2} - \frac{|b-c|}{2}\right)\right|}{2},$$ which isn't intuitively the minimum of three numbers, and isn't even symmetrical in the variables, even though its output is. Is there some nicer way of expressing this function?

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You could use the unit step function or Iversonian brackets, but I suppose the resulting expression cannot be considered in any way nice... – J. M. Dec 7 '10 at 4:42
Could you list the original reference for the expressions? – user13769 Jul 26 '11 at 1:01
$\min\{a,b,c\}$ is nice. – Raphael Jul 26 '11 at 5:14
Please note that it is much nicer (IMHO) to do it the other way around, i.e., to express absolute value in terms of min/max: $\;|a| = a \max -a\;$, and $\;a \min b = - (-a \max -b)\;$. min/max have nice properties like associativity, $\;+\;$ distributes over them, etc. See my answer for an example. – Marnix Klooster Dec 10 '13 at 5:35
up vote 22 down vote accepted

First, define $$ \Delta=|a-b|+|b-c|+|c-a|\newcommand{\Mu}{\mathrm{M}}\tag{1} $$ It is somewhat intuitive that $$ \frac{\Delta}{2}=\max(a,b,c)-\min(a,b,c)\tag{2} $$ For example, if $a\ge b\ge c$ then $|a-b|+|b-c|+|c-a|=2a-2c$.

Next, define $$ \Sigma=a\left(1-\frac{|b-c|}{\Delta}\right)+b\left(1-\frac{|c-a|}{\Delta}\right)+c\left(1-\frac{|a-b|}{\Delta}\right)\tag{3} $$ Again, if $a\ge b\ge c$, then $$ \begin{align} \Sigma &=a\left(\frac{2a-b-c}{2(a-c)}\right)+b\left(\frac{a-c}{2(a-c)}\right)+c\left(\frac{a+b-2c}{2(a-c)}\right)\\ &=a+c \end{align} $$ Thus, considering the symmetry of $(3)$, it is evident that $$ \Sigma=\max(a,b,c)+\min(a,b,c)\tag{4} $$ Combining $(2)$ and $(4)$ yields $$ \max(a,b,c)=\frac{\Sigma}{2}+\frac{\Delta}{4}\tag{5} $$ and $$ \min(a,b,c)=\frac{\Sigma}{2}-\frac{\Delta}{4}\tag{6} $$ At least $(5)$ and $(6)$ are symmetric in $a$, $b$, and $c$ since $(1)$ and $(3)$ are. That is, $$ \begin{align} \max(a,b,c) &=\frac{a}{2}\left(\frac{|c-a|+|a-b|}{|a-b|+|b-c|+|c-a|}\right)\\ &+\frac{b}{2}\left(\frac{|a-b|+|b-c|}{|a-b|+|b-c|+|c-a|}\right)\\ &+\frac{c}{2}\left(\frac{|b-c|+|c-a|}{|a-b|+|b-c|+|c-a|}\right)\\ &+\frac{|a-b|+|b-c|+|c-a|}{4}\tag{7} \end{align} $$ and $$ \begin{align} \min(a,b,c) &=\frac{a}{2}\left(\frac{|c-a|+|a-b|}{|a-b|+|b-c|+|c-a|}\right)\\ &+\frac{b}{2}\left(\frac{|a-b|+|b-c|}{|a-b|+|b-c|+|c-a|}\right)\\ &+\frac{c}{2}\left(\frac{|b-c|+|c-a|}{|a-b|+|b-c|+|c-a|}\right)\\ &-\frac{|a-b|+|b-c|+|c-a|}{4}\tag{8} \end{align} $$

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Is it possible to generalize this idea to rewrite $\min(a_1, ... a_n)$ ? Although it would be surely awful as equation I'm interested ;-) – Thomas Produit Jan 29 '14 at 19:11
If $a$, $b$ and $c$ are vectors for the vertices of a triangle then $\Sigma/2$ is its Spieker Center and $\Delta/4$ is its semiperimeter. Is there any significance to the circle about $\Sigma/2$ with radius $\Delta/4$? – Oscar Cunningham Jun 30 '14 at 16:36
GeoGebra suggests that this circle doesn't (as one might hope) always go through two vertices of the triangle. When the vertices are not collinear it goes through none of the points, instead going around the triangle on the outside of all the vertices. – Oscar Cunningham Jun 30 '14 at 17:02

This probably isn't what you were thinking of, but if $a_1, ... a_n$ are non-negative, then

$$\min(a_1, ... a_n) = \lim_{k \to -\infty} \sqrt[k]{a_1^k + ... + a_n^k}$$


$$\max(a_1, ... a_n) = \lim_{k \to \infty} \sqrt[k]{a_1^k + ... + a_n^k}.$$

There are several applications of these identities (at least the second one), e.g. to functional analysis. They are also related to the way in which tropical arithmetic arises as a "limit" of ordinary arithmetic.

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Here is a hint why there is no simple such formula:

In the case of two variables $a_i$ they are the zeros of the polynomial $$p(x):=(x-a_1)(x-a_2)=x^2 -(a_1+a_2)x + a_1 a_2\ .$$ Therefore by the formula for quadratic equations we have $$\min(a_1, a_2)\ =\ {a_1+a_2-\sqrt{(a_1+a_2)^2 -4a_1 a_2}\over 2} ={a_1+a_2\over2}-{|a_1-a_2|\over 2}\ .$$

Using the same idea with three variables $a_i$ we would have to look at the third degree polynomial $$q(x):=(x-a_1)(x-a_2)(x-a_3)$$ which has three real roots. Therefore we are in the "casus irreducibilis" of Cardano's formula, which can only be solved via complex numbers. Even if you would write everything out, you couldn't decide "by inspection" which root is the smallest.

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That's good stuff. – The Chaz 2.0 May 29 '13 at 17:30

If you want a symmetric expression, you can take $\frac13 (\min(a,\min(b,c))+\min(b,\min(a,c))+\min(c,\min(a,b)))$. But if you rewrite it using that absolute-value trick, I still don't think it gives something that's "intuitively the minimum", I'm afraid...

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