Find the minimum of several functions

Question

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
}

Your help is appreciated,

Reid

Answer 1

Try this

$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.