Intersection Bodies of Polytopes: Translations and Convexity

We continue the study of intersection bodies of polytopes, focusing on the behavior of $IP$ under translations of $P$. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of $I(P+t)$ can be extended to polynomials in variables $t\in {\mathbb{R}}^d$ within each region of the arrangement. In dimension $2$, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions.