Let $K\_1, K\_2, \ldots K\_n$ be convex bodies in $R^d$. Assume that for any index set $I$, $\cap\_{i \in I} K\_i$ is not empty and is not properly contained in any body $K\_i$ for $i\in I$.


Is it true that $\cap\_{i \in I} \partial K\_i$ is a disjoint union of topological (or even better PL) spheres?



Is the intersection of boundaries of convex bodies a topological sphere?
A counterexample is as follows:


Let $K\_1=\{x,y,z,w|x^2+y^2\leq 1\}$ and $K\_2=\{x,y,z,w|z^2+w^2\leq 1\}$. The boundaries are just what you get when you replace the inequality with an equality, so their intersection is a torus, $x^2+y^2=1$, $z^2+w^2=1$.


Then the interiors intersect, for instance at the origin, fulfilling Patricia Hersh's condition. The boundaries are topologically transverse, fulfilling Richard Kent's condition. I cannot think of any additional reasonable condition that would disallow this counterexample.


