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∈ Rᵈ t ∈ 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.