# Max function with absolute values for arbitrary number of arguments

The maximum between two numbers $$x$$ and $$y$$ can be easily written as $$max(x,y) = \frac12\left(x+y +|x-y|\right).$$ We can obviously generalize this to any number of arguments as $$max(x_1,\dots,x_n) = max(x_1, max(... ,max(x_{n-1},x_n)...)) = \text{ a mess with absolute values}$$ but I do not like this. I would like to have a nicely written formula and the formula should look the same if we shuffle $$x_1,\dots,x_n$$.

Why? You might ask? I'm doing some numerical computations where I need to smooth out the $$max$$ function, I just trivially smooth out the absolute value: $$max_{\varepsilon}(x,y) = \frac12\left(x+y +|x-y|_{\varepsilon}\right) = \frac12\left(x+y +\sqrt{ (x-y)^2 + \varepsilon^2} - \varepsilon\right).$$ However, I actually need to smooth out the $$max$$ function for arbitrary number of arguments. The basic requirement on $$max_\varepsilon(x_1,\dots, x_n)$$ is that it is invariant under any permutation of its arguments. Therefore if I have $$max(x_1,\dots,x_n)$$ written out with absolute values and the expression is symmetrical in $$x_1,\dots,x_n$$ then I can just replace $$|\cdot|$$ with smoothed version of absolute value $$|\cdot |_\varepsilon$$ and I get the desired result.

Thus, is there an expression with just absolute values which is symmetrical in $$x_1,\dots,x_n$$ and when evaluated it yields the maximum?

• Would you be happy with en.wikipedia.org/wiki/Smooth_maximum ? – Ru Hasha Jan 18 at 15:02
• Cool! That should solve my problem, but I'm still curious about the question I have posted. – tom Jan 18 at 18:09