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A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$.

Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that calculates the combination of all $k$ polyhedra $ P_1\cup P_2 \cup \dotsb \cup P_k$, and gives the final shape's vertices and facets (extreme points and faces)?

Or is this an ongoing research topic?

After a search online, I found most algorithms are focusing on 2D and 3D polyhedra. CGAL has implementations on boolean operations of 2D and 3D Nef Polyhedra (https://doc.cgal.org/latest/Nef_2/index.html and https://doc.cgal.org/latest/Nef_3/index.html).

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    $\begingroup$ You need to rewrite this question. At this time, it makes no sense. $\endgroup$
    – Wlod AA
    Commented Oct 19, 2022 at 23:19
  • $\begingroup$ For example, doc.sagemath.org/html/en/reference/spkg/polylib.html $\endgroup$
    – Max Alekseyev
    Commented Oct 19, 2022 at 23:20
  • $\begingroup$ @MaxAlekseyev I took a look at this library, it can calculate the union of polyhedra, but the union is just a list of all given polyhedra, not in terms of the final shape's vertices and facets. $\endgroup$
    – Robin Lee
    Commented Oct 20, 2022 at 0:24
  • $\begingroup$ @WlodAA Could you let me know which part is comfusing? $\endgroup$
    – Robin Lee
    Commented Oct 20, 2022 at 0:25
  • $\begingroup$ @RobinLee: I don't think it's "just a list". Here is a more detailed page about this library: icps.u-strasbg.fr/polylib $\endgroup$
    – Max Alekseyev
    Commented Oct 20, 2022 at 0:51

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