computing the boundary of a union of polytopes Let $P_1,\dots ,P_m\subset \mathbb{R}^n$ be $m<\infty$ convex polytopes in $\mathbb{R}^n$, and $U:=\bigcup_{j} P_j$ their set-theoretic union. What algorithms are known for computing the boundary $\partial U$ of $U$?
Note that $U$ would not always be convex, might have a nontrivial fundamental group, etc.
A naive algorithm I can think about would start from $\partial P_1$, intersect each facet of $P_2$ with it, obtaining $\partial (P_1\cup P_2)$, etc. 
A less naive point of view might involve working with measures supported on $P_j$'s and inclusion/exclusion in some way.
Any references on this sort of questions?
 A: Not a direct answer, but may help with: "Any references on this sort of questions?"
Computing the convex hull of the union of two polytopes $P_1 \cup P_2$
is NP-hard:

Tiwary, Hans Raj. "On the hardness of computing intersection, union and Minkowski sum of polytopes." Discrete & Computational Geometry 40, no. 3 (2008): 469-479. Author preliminary PDF download.

But the computation is polynomial for special polytopes:

Balas, Egon. "On the convex hull of the union of certain polyhedra." Operations Research Letters 7, no. 6 (1988): 279-283. Journal link.
Conforti, Michele, Marco Di Summa, and Yuri Faenza. "Balas formulation for the union of polytopes is optimal." arXiv:1711.00891 (2017).

 Added.
Specializing to $\mathbb{R}^3$, there is an algorithm
that constructs the boundary of the union,
which runs in $O(k^3 + kn \log k \log n)$ expected time,
where $k$ is the number of polyhedra and $n$ is the 
total number of faces:


Aronov, Boris, Micha Sharir, and Boaz Tagansky. "The union of convex polyhedra in three dimensions." SIAM Journal on Computing 26, no. 6 (1997): 1670-1688.
  Journal link

            
  

            
  
  Aronov, Sharir, Tagansky.
  


Subsequently there was a specialized extension to $\mathbb{R}^4$:

Aronov, Boris, Alon Efrat, Vladlen Koltun, and Micha Sharir. "On the union of κ-round objects in three and four dimensions." Discrete & Computational Geometry 36, no. 4 (2006): 511-526. Journal link.

