Connected components of the difference of convex bodies

Question

Let $K \subset \mathbb{R}^d$ be a closed symmetric convex body and consider the difference $D = K \setminus B$ where $B$ is a ball centered at the origin. Is there a bound on the number of connected components of $D$, perhaps in terms of the number of vertices of $K$? It seems for $K$ an ellipsoid you can have at most two components, while for $K$ a regular polytope there can be as many as the number of vertices.