Simple topological question on taking complements inside a simplex

2010-09-16T15:30:43Z

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

$\cup_{i \in I \subseteq [d]} P_i$ is a polytope,
$\cup_{i \in [d]} P_i$ is a simplex,
their interior (denoted by $P_i^{o}$) is disjoint,
$P_i$ 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').

Simple topological question on taking complements inside a simplex - MathOverflow

Simple topological question on taking complements inside a simplex