Can the convex hull of a set of vectors $x_1,x_2,...,x_n$ such that $x_i\in\mathbb{R}^4$ for each $i\in N$ be a polyhedron in $\mathbb{R}^3$? I think that the answer is affirmative, but I am not sure. Other than that, let me provide particular example. Consider the convex hull given by the following vectors: \begin{equation} Co(X)=Co\{(1,0,1,1),(0,0,2,1),(0,0,1,2),(0,1,1,1),(1,1,1,0),(1,1,0,1),(1,2,0,0)\} \end{equation}
Then, my specific questions are:
- Can the convex hull $Co(X)$ be represented as a polyhedron in $\mathbb{R}^3$?
- If the answer to the previous question is affirmative, how do we do so? In other words, how do we find equivalent vertices in $\mathbb{R}^3$ that allow us to graphically represent $Co(X)$?
Thank you all very much in advanced for your time.
PS: If possible, provide some calculations on the required steps or else try to provide some useful resources.
If what you're asking is whether that convex hull is three dimensional (lives in a three dimensional affine subset of four space) then the way to find out is to subtract one point from all the rest and determine whether the resulting set of $n-1$ points is independent. If it is, the original polytope is four dimensional. If not, its dimension is less: the dimension of that span.
I haven't worked out your particular example.