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 \(I(P+t)\) $I(P+t)$ can be extended to polynomials in variables \(t\in \mathbb {R}^d\) $t\in {\mathbb{R}}^d$ 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.