Permutahedron edge

Question

Let's suppose we have the permutahedron (1,2,3,4) and all the possible permutations . I try to find an edge of this polytope . I try to find a hyperplane a1*x1+a2*x2+a3*x3+a4*x4 = b, such that (1,3,2,4) and (1,4,2,3) passing through the hyperplane i.e a1*(1) + a2*(3) + a3*(2) + a4*(4) = b
a1*(1) + a2*(4) + a3*(2) + a4*(3) = b But we want for all the other permutations to be a1*x1 + a2*x2 + a3*x3 + a4*x4 > b or a1*x1 + a2*x2 + a3*x3 + a4*x4 < b and of course not for some permutations < b and for other >b . Sorry for my english.

Answer 1

To define an edge in four dimensions requires three intersecting hyperplanes. One hyperplane is the one in which the entire permutahedron lives:

$$ a_1+a_2+a_3+a_4=10\;. $$

To find the other two, look at this Wikipedia illustration. The edge between $1324$ and $1423$ (coloured in red) is the intersection of the faces that $1324$ and $1423$ form with $2413$ and $2314$ (a square) and with $1432$, $1342$, $1243$ and $1234$ (a regular hexagon). You could use generalized cross products to find vectors that are normal to these faces, but it's also not too difficult to guess hyperplanes that define them (together with the above hyperplane for the entire permutahedron), for instance,

$$ a_1-a_2+a_3-a_4=-4 $$

for the square and

$$ a_1=1 $$

for the regular hexagon.