# Simple topological question on taking complements inside a simplex

We would like to know if the following claim is true: (If you don't know the definition of a tropical hyperplane, then please consider the case when d=3)

Let $P_1,\cdots,P_d$ be full dimensional (possibly non-convex) polytopes in $\mathbb{R}^{d-1}$ such that

1. $\cup_{i \in I \subseteq [d]} P_i$ is a polytope,
2. $\cup_{i \in [d]} P_i$ is a simplex,
3. their interior (denoted by $P_i^{o}$) is disjoint,
4. $P_i$ contains only one vertex of the simplex.

Then in the interior of the simplex, the complement of $\cup_{i \in [d]} P_i^{o}$ is PL-homeomorphic to a tropical hyperplane (When d=3, this looks like a 'Y').

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I am trying to guess what you mean by "polytope". Maybe you mean a PL-space that is PL-homeomorphic to a simplex? – Tom Goodwillie Sep 16 '10 at 16:11
Is condition 1 for all I? – Thierry Zell Sep 16 '10 at 17:15
(I suppose that the answer to both the preceding questions is "yes".) You don't know if the intersection of the $P_i$ is also a polytope, right? This seems a crucial point. – Bruno Martelli Sep 16 '10 at 17:47
Thank you for the comments. The answer for both questions is a yes, and we really don't know how the intersections look like yet. – Suho Oh Sep 16 '10 at 18:06