Equation of a rectangle in $R^{n}$

Question

I have 4 points in $R^{8}$, which make a rectangle with length $2$ and width $sqrt(2)$. The vertices are:

$(0,0,1,1,1,1,0,0),(1,0,0,0,0,1,1,1),(0,0,0,1,1,1,1,0),(0,0,0,0,1,1,1,1)$

what is the equation of this rectangle in $R^{8}&?

In general I am wondering if we are given 4 points in $R^{n}$ what is the equation of that rectangle?! But if you just could solve my question in $R^{8}$ that’s fine! I will apperatiate !

Answer 1

Hint: In your example you said $a<x<b$, $c<y<d$, but how would you find the equation of a slanted rectangle?

Answer 2

Wrong, those 4 points do span a 3D subspace.
In fact those are 4 vertices from a cube of edge length $\sqrt{2}$.

$\overline{AC} = \overline{CD} = \overline{DB} = \sqrt{2}$ (: edges)
$\overline{AD} = \overline{CB} = \sqrt{4}$ (: square diagonals)
$\overline{AB} = \sqrt{6}$ (: body diagonal)

--- rk