Let $d$ be an integer. Let $A,B \subseteq \mathbb R^d$ be two sets homeomorphic to an open $d$-ball such that their intersection is again homeomorphic to an open $d$-ball. Does it follow that their union is homeomorphic to an open $d$-ball?
The motivation comes from nerve type theorems. For instance it follows in the above-mentioned case that $A \cup B$ has to be nullhomotopic.
In more general setting, let $A_1, \dots, A_n \subseteq \mathbb R^d$ be a collection of sets homeomorphic to an open $d$-ball such that the intersection of every subcollection is either empty or homeomorphic to an open $d$-ball. Moreover, let us assume that it is determined which subcollections are supposed to have a nonempty intersection. This data can be "stored" as a simplicial $K$ with vertices $A_1, \dots, A_n$ and whose faces are exactly that subcollections which have a nonempty intersection.
By standard nerve theorems, $K$ has to be homotopy equivalent to $A_1 \cup \cdots \cup A_k$; however I wonder whether anyone is aware of any "homeomorphism-type" nerve theorem:
Let $A_1, \dots, A_n \subseteq \mathbb R^d$ and $K$ be described as above. Does it follow that the homeomorphism type of $A_1 \cup \cdots \cup A_n$ is determined by $K$? Does it follow if $K$ is at most $d$-dimensional?
Even if the answer to the question above is negative, an interesting specific case occurs when $A_1, \dots, A_n$ are assumed to be convex. (Note that the answer to first question is positive in this case, since $A \cup B$ is star-convex.)
I came to these questions when I was considering a certain algorithmic result on the collections of sets as described above. With a coauthor, we finally circumvent these questions (it showed up to be more convenient). However, I would be still very curious about the answers.