Intersection Bodies of Polytopes: Translations and Convexity

Abstract

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 ∈ R d within each region of the arrangement. Establishing the convexity space as the set of translations such that I(P + t) is convex, we fully characterize it for two-dimensional polytopes and partially characterize it for higher dimensions, revealing unexpected finite behavior in the two-dimensional case and for the d-dimensional cube.