TITLE: Is the intersection of boundaries of convex bodies a topological sphere?
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: To get a "YES" answer, you have to assume that 
at any point of $p\in\partial K_i\cap \partial K_j$
any two supporting hyperplanes to $K_i$ and $K_j$ 
have angle $> \tfrac{\pi}2$.
The proof is by induction on $n$.
WLOG we may assume that all $\partial K_i$ are smooth.
Assume $S_{n-1}=\partial K_1\cap \partial K_2\cap \dots\cap K_{n-1}$ is a sphere.
Note that $f=\mathop{\rm dist}_{\partial K_n}$ is a concave function on $S_{n-1}\cap K_n$,
Perturb $f$ so it become smooth.
The function has one maximum point and by Morse Lemma the level set 
$S_n=f^{-1}(0)=\partial K_1\cap \partial K_2\cap \dots\cap K_{n}$ is a sphere.