# Find the minimum of several functions

I have 6 functions of four variables $f_i: \mathbb{R}^4 \rightarrow \mathbb{R},\, i \in 1..6$

For each function $f_j$ I want to find the subset of $\mathbb{R}^4$, called $S_j$, for which it is less than all the other functions

i.e.

$f_j(w,x,y,z) \le f_i(w,x,y,z) \; \forall i \in 1..6 \; \forall (w,x,y,z) \in S_j$

In particular the functions I'm working with are as follows:

{
1/2 (-x - y + Sqrt[4 w^2 + x^2 - 2 x y + y^2]),
1/2 (-x - y - Sqrt[4 w^2 + x^2 - 2 x y + y^2]),
-z,
x + y - Sqrt[4 w^2 + x^2 - 2 x y + y^2],
x + y + Sqrt[4 w^2 + x^2 - 2 x y + y^2],
2 z
}


$Assumptions = {w, x, y, z} \[Element] Reals; f = {1/2 (-x - y + Sqrt[4 w^2 + x^2 - 2 x y + y^2]), 1/2 (-x - y - Sqrt[4 w^2 + x^2 - 2 x y + y^2]), -z, x + y - Sqrt[4 w^2 + x^2 - 2 x y + y^2], x + y + Sqrt[4 w^2 + x^2 - 2 x y + y^2], 2 z}; Table[c = f[[i]]; d = Drop[f, {i}]; Simplify[Reduce[And@@Map[c<=#&, d], {w, x, y, z}]], {i, 1, 6}]  That extracts each of your six expressions, uses those to construct the inequalities with the remaining five and gives that to Reduce to find your solutions. • Thank you @Bill! I was trying to construct a piece-wise function of 5 arguments (index$ i \in 1..6\$ and the other four), and use Minimize on that to no avail. This is much better. Jul 13, 2016 at 20:09