Given is a set of points $S = \{s_1,...,s_N\in\mathbb{R}^n\}$ and a plane $H_1$ with $H_1^+ = \{ x \in \mathbb{R}^n ~\vert~ n_1^Tx - b_1 \geq 0\}$ being its positive half-space, such that $\forall s\in S : s \in H_1^+$ and $\frac{1}{N}\sum_{s\in S} (n_1^Ts - b_1)$ is minimized. Such a plane can be found by solving an LP of the form $$ \min F^T z\\ \text{subject to} \\ \forall s\in S: \begin{pmatrix}-s^T&1\end{pmatrix}z \leq 0, \\ -F^T z \leq -1, $$ where $F = \begin{pmatrix}\frac{1}{N}\sum_{s\in S} s\\-1\end{pmatrix}$.
That works. Now however I want to find an additional plane $H_2$ with normal $n_2$ and bias $b_2$, which also satisfies the above criteria and it potentially intersects, but is not equal to $H_1$.
My question is how to formulate this as an LP in order to find $n_2$ and $b_2$.
More contextually my problem is about finding planes $H_1,...,H_p$ which form a polytope $P = \bigcap_{i=1}^p H_i^+$, such that $\text{conv}S \subseteq P$. My approach (and hence the question) is to start with an initial plane $H_1$ and solving extra LPs for each new plane, until a bounding polytope is found.
Any advice or perhaps an alternative approach would be greatly appreciated. I admit that my knowledge of linear programming is quite limited but I am slowly catching up.
EDIT: To clarify the general problem a bit, I am not trying to find the convex hull, but rather an over-approximation of it given a fixed number of $p = n+k$ faces.