Intersection Bodies of Polytopes

TL;DR

This paper studies the intersection bodies of convex polytopes, demonstrating their semialgebraic nature, providing algorithms for their computation, and analyzing their algebraic boundaries and degrees.

Contribution

It introduces the first comprehensive analysis of intersection bodies of polytopes using combinatorics and algebraic geometry, including algorithms and boundary degree bounds.

Findings

Intersection bodies of polytopes are semialgebraic sets.

An algorithm for computing intersection bodies is provided.

Upper bounds for the degrees of algebraic boundary components are established.

Abstract

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for its computation. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for the degree of these components.