I have the following problem:

I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq d_i \quad \forall (n_i,d_i) \in S $. Now I have a "joining" hyperplane $(n_{k+1},d_{k+1})$ and I want to know if this hyperplane "modifies the shape" of the convex hull and in that case, which hyperplanes of $S \bigcup (n_{k+1},d_{k+1})$ are not necessary anymore because they become redundant.

Trivial example with one dimension:

My convex hull is described by the inequality $ 3 \leq x \leq 5$ so

$S = [(1,3),(-1,-5)]$

The joining hyperplane is the inequality $ x \geq 4$ so the resulting convex hull should be $ 4 \leq x \leq 5$

$S = [(1,4),(-1,-5)]$

returning $(1,3)$.

Now I would like the same thing generalized for n-dimensions. I can get algorithms till 3 dimensions but they are not generalized.

Do you have any hints or pointers on how I can find a solution to this problem?

p.s. I apologize for the sloppy description, I am not a mathematician. Please feel free to ask for more details.

Kind regards.