Let $K_1, \ldots, K_n \subset \mathbb{R}^n$ be some convex bodies (i.e., compact with nonempty interior) such that for each subset $I \subset \{1,\ldots,n\}$ the set
$$\left( \bigcap_{i \in I} K_i \right) \setminus \left( \bigcup_{i \not\in I} K_i \right)$$
has nonempty interior. Does it follow that the intersection of the boundaries $\bigcap_{i \in I} \partial K_i$ is nonempty?
It seems to be true for $n \leq 3$ although I don't have a formal proof.
If the question is whether the intersection of the boundaries of all the sets $K_i$ is nonempty, the answer is no for $n=3$.
Let $K_1$ be the unit ball with center $(0,0,0)$, $K_2$ the unit ball with center $(0,0,1)$, and $K_3$ a thin cylinder of radius $0.1$ around the $z$-axis, with bases at heights $-2$ and $2$. The condition on intersections is clearly satisfied.
The boundaries of $K_1$ and $K_2$ intersect in a circle $C$ of radius $\sqrt{3}/2 > 0.5$, so $K_3$ is disjoint with $C$.
