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.



Common boundary point of convex bodies
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 $C$. 


