Intersection Bodies

Leaving the realm of polytopes and convex bodies.

published article: SpringerLink
arXiv: 2110.05996
Slides: slides.pdf

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.

Here is a video animation that I made to explain the construction of intersection bodies: [video]

Our MathRepo page contains lots of additional supplementary material:

  • An implementation of an algorithm in SageMath to compute interseciton bodies of polytopes
  • A step by step explanation of the algorithm with the example of the cube
  • A gallery of 3d models that show how different intersection bodies can look like

Here are two of the 3d models that you can find in the gallery. Click here so see them all!

(We are 3d models, you can rotate us)