Nice expression for minimum of three variables?

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

min(a,b)=a+b2|ab|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:
a+(b+c2|bc|2)2|a(b+c2|bc|2)|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?

Answer

This probably isn’t what you were thinking of, but if a1,...an are non-negative, then

min(a1,...an)=limkkak1+...+akn

whereas

max(a1,...an)=limkkak1+...+akn.

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.

Attribution
Source : Link , Question Author : Oscar Cunningham , Answer Author : Qiaochu Yuan

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