I have a set of points in $\mathbb{R}^n$ and I can use some convex hull algorithm to find the convex hull of this set and the corresponding extreme points (vertices). My goal is to find the supporting hyperplane that contains a point $p$ where all elements are the same, i.e., $p=[p_1,p_2,...,p_n]$ and $p_1=p_2=...=p_n$. I can guarantee that if we create a ray of $p_1=p_2=...=p_n$ in $\mathbb{R}^n$, this ray must pass the boundary of the convex hull.
I know there is a facet enumeration algorithm that we can find a H-representation given a V-representation of the convex hull. However, I just want to find the supporting hyperplane that contains the point $p$. Therefore, I am wondering if it is possible to just find such a hyperplane instead of finding all supporting hyperplanes of the convex hull.
Edit: The set of points $A$ is the convex hull of the extreme points of the unions over some hyperrectangles. Specifically, it can be described as follows:
$A=\text{Conv}(\text{extr}(A_1\cup A_2\cup A_3))$ where $A_1,A_2,A_3$ are the collection of some points satisfying some conditions. Specifically, for any $i\in\{1,2,3\}$, $A_i=\{a_{i,1},...,a_{i,n}:0\leq a_{i,j}\leq c_i\,\forall\, j\in\{1,...,n\}\}$ where $c_i$ is a positive constant for each $i$ and $c_i\neq c_j$ for any $i\neq j$ is allowed.
Edit: I conjecture that the possible steps may be
- Find the vertices that can form a hyperplane that contains $p$.
- Use these vertices to create a hyperplane.
In the first step, we may investigate the amplitude of each coordinate of each vertex. In the second step, we may solve some linear algebra to find the hyperplane formed by the set found in the first step.