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Let $p_1, \dots, p_k$ be a sequence of convex polytopes such that $p_i$ and $p_{i+1}$ intersect somewhere on their boundaries but have disjoint interiors. We can think of these polytopes as forming a corridor: $$ C = \bigcup_{i=1}^k p_i. $$

We want to find a new sequence of convex polytopes $p_1', \dots, p_k'$ such that:

  • $p_i \subseteq p_i'$
  • $p_i' \subseteq C$
  • the volume of $p_i'$ is maximal

In other words, we want to find a new corridor of polytopes whose union is the same corridor, but who overlap as much as possible.

Is there a name for this problem, and if so is there any known algorithm to compute the expanded polytopes?

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