Simple topological question on taking complements inside a simplex - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:43:15Z http://mathoverflow.net/feeds/question/38982 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38982/simple-topological-question-on-taking-complements-inside-a-simplex Simple topological question on taking complements inside a simplex Suho Oh 2010-09-16T15:30:43Z 2010-09-16T15:30:43Z <p>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)</p> <p>Let $P_1,\cdots,P_d$ be full dimensional (possibly non-convex) polytopes in $\mathbb{R}^{d-1}$ such that</p> <ol> <li>$\cup_{i \in I \subseteq [d]} P_i$ is a polytope,</li> <li>$\cup_{i \in [d]} P_i$ is a simplex,</li> <li>their interior (denoted by $P_i^{o}$) is disjoint,</li> <li>$P_i$ contains only one vertex of the simplex.</li> </ol> <p>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').</p>