# How to plot a convex polytope in 3D when the polytope is given by vectors in $\mathbb{R}^4$?

Suppose that I want to plot the convex hull of the following set of vectors: namely, $$Co(X)=Co\{(1,0,1,1),(0,0,2,1),(0,0,1,2),(0,1,1,1),(1,1,1,0),(1,1,0,1),(1,2,0,0)\}$$ Because the matrix given by the original set of vectors has rank $4$, its convex hull does not lie in a three-dimensional space. However, the matrix given by my original vectors can be translated by subtracting the first vector in such a manner that its convex hull is three-dimensional. In other words, the following convex hull is three-dimensional because the translated matrix has rank $3$ and can therefore be plotted: $$Co(X^{\prime})=Co\{(0, 0, 0, 0), (-1, 0, 1, 0), (-1, 0, 0, 1), (-1, 1, 0, 0), (0, 1, 0, -1), (0, 1, -1, 0), (0, 2, -1, -1)\}$$

Hence, my question is simple: can anybody provide some code that plots the convex hull $Co(X^{\prime})$?

PS: In case somebody wants a little bit of background to see where this question comes from, see this question and this other one.

Just project all points into 3D plane.

mat = {{-1, 0, 1, 0}, {-1, 0, 0, 1}, {-1, 1, 0, 0}, {0, 1, 0, -1}, {0,
1, -1, 0}, {0, 2, -1, -1}}
ort = NullSpace[{{-1, 0, 1, 0}, {-1, 0, 0, 1}, {-1, 1, 0, 0}, {0, 1,
0, -1}, {0, 1, -1, 0}, {0, 2, -1, -1}}]
ns = NullSpace[ort]
data3d = mat.Transpose[ns]

Needs["TetGenLink"]
{pts, surface} = TetGenConvexHull[data3d];
Graphics3D[GraphicsComplex[pts, Polygon[surface]]]
`
• Thank you very much for your answer. Could you please tell me how to get a ball in the corresponding vector $\kappa=(\frac{1}{2},\frac{1}{2},\frac{7}{6},\frac{5}{6})$, please? Feb 6, 2018 at 18:13
• @Héctor how to get a what? Feb 6, 2018 at 18:35
• A ball (or a circumference, or a dot or a square or some symbol) indicating that where the vector $\kappa=(\frac{1}{2},\frac{1}{2},\frac{7}{6},\frac{5}{6})$ is located in the plot. Feb 6, 2018 at 18:44
• @Héctor it's not in that 3D plane. It's projection has coordinates {1/3, 2/3, 0}. Feb 6, 2018 at 18:52
• Exactly! And how would you plot that $\kappa^{\prime}=(\frac{1}{3},\frac{2}{3},0)$ in the figure? Feb 6, 2018 at 18:58