Let us consider a set of $N$ vectors, $\mathbf{h}_1, \mathbf{h}_2, \cdots, \mathbf{h}_N$, such that $\mathbf{h}_i \in \mathbb{R}^{M}$, $\forall i$, with $M < N$.
Let us also consider the space $\mathcal{X}$ obtained from bouded linear combinantions of such vectors, i.e.
$$\mathcal{X} = \{\mathbf{x} \;|\; \mathbf{x} = \sum_{i=1}^{N}a_i\mathbf{h}_i, a_i \in [-A,A], \forall i\}$$
I know that $\mathcal{X}$ corresponds to the convex hull of the vertices of such a set. Also, the vertices are the points obtained when the coefficients are saturated.
Question: Is it possible to estimate the set of vectors $\mathbf{h}_1, \mathbf{h}_2, \cdots, \mathbf{h}_N$ from the convex hull (e.g. by using the vertices of the convex hull or any other informaion) ?
For $M=2$, I could relate the vectors $\mathbf{h}_1, \mathbf{h}_2, \cdots, \mathbf{h}_N$ to the sides of the convex hull. However, I could not prove that such a relation exists for higher dimensions ($M \geq 3$).
$\mathcal X$ is not the convex hull of the set of $\mathbf h_i$, but rather of the set $H = \{\pm A\mathbf h_i \mid i = 1,2, \dots, N\}$. Further, it is not always necssary for all the $\mathbf h_i$ to be included. to get the same convex hull. SOme of the $A\mathbf h_i, -A\mathbf h_i$ may fall within the convex hull of the others, in which case, their inclusion will not change that convex hull.
We can divide $H$ into two sets: $H_1$ are the vectors in $H$ that cannot be removed without changing the convex hull, while $H_2$ are those that can be removed without changing it. The vectors in $H_1$ will all be vertices of $\mathcal X$, and thus can be identified from $\mathcal X$ itself. But note that if $\mathbf v \in \mathcal X$, then so is $-\bf v$. That is, $\mathcal X$ is symmetric through the origin. So if $\bf v$ is a vertex of $\cal X$, so is $-v$. Thus $A\mathbf h_i \in H_1$ if and only if $-A\mathbf h_i \in H_1$. However, determining the $\mathbf h_i$ with $\pm A\mathbf h_i \in H_1$ would require knowing the value of $A$ as well.
On the other hand, the $\mathbf h_i$ with $\pm A\mathbf h_i \in H_2$ cannot be identified at all from $\cal X$. Any vectors in $\cal X$ could be substituted for them without any change in $\cal X$.