Let $P$ be a polytope of dimension $n$, and let $\mathcal{C}$ be a polyhedral subdivision of $P$. That means that $\mathcal{C}$ is a finite collection of polytopes whose union is $P$, such that the intersection of any two elements of $\mathcal{C}$ is a common face of both and is itself contained in $\mathcal{C}$.
Let $U$ be a convex open set that is contained in the interior of $P$, and let $A$ be the union of the relative interiors of all elements of $\mathcal{C}$ that touch $U$. That is, $$A := \bigcup_{\substack{Q\in \mathcal{C}\\ Q\cap U\neq \emptyset}} \operatorname{Int}(Q).$$ Note that $A$ is an open subset of the interior of $P$.
I would like to prove that $A$ is homeomorphic to a ball. I would be just as happy to prove the following slightly weaker statement: $$H_k(P,P\setminus A; \mathbb{Q}) \cong H_k(S^n; \mathbb{Q})$$ for all $k$.
