# Intersection Bodies of Polytopes: Translations and Convexity

## TL;DR

This paper investigates how intersection bodies of polytopes change under translations, providing polynomial descriptions and characterizations of convexity in two dimensions, with partial results in higher dimensions.

## Contribution

It introduces an affine hyperplane arrangement framework and characterizes convexity of intersection bodies of polygons, advancing understanding of geometric transformations.

## Key findings

- Polynomials describing $I(P+t)$ extend within each region of the hyperplane arrangement.
- Full characterization of polygons with convex intersection bodies in 2D.
- Partial characterization results for higher dimensions.

## 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 $t\in \mathbb{R}^d$ 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.