Let $K_1, K_2, \ldots K_n$ be convex bodies in $R^d$. (Assume some generality condition, so that in particular, for an index set $I$, $dim(\cap_{i \in I} \partial K_i)=d-|I|$).

Consider an index set $I$, is it true that $\cap_{i \in I} \partial K_i$ is a disjoint union of topological (or even better PL) spheres?