Positive Geometry of Polytopes and Polypols

TL;DR

This paper introduces positive geometries and their canonical forms, focusing on polytopes and polypols, and compares different definitions while providing computational examples and exercises.

Contribution

It presents a comprehensive overview of positive geometries, including original and recent definitions, and computes canonical forms for convex polytopes and quasi-regular polypols.

Findings

Computed canonical forms for convex polytopes.

Analyzed quasi-regular polypols as nonlinear polygon generalizations.

Compared different definitions of positive geometries.

Abstract

These are lecture notes supporting a minicourse taught at the Summer School in Total Positivity and Quantum Field Theory at CMSA Harvard in June 2025. We give an introduction to positive geometries and their canonical forms. We present the original definition by Arkani-Hamed, Bai and Lam, and a more recent definition suggested by work of Brown and Dupont. We compute canonical forms of convex polytopes and of quasi-regular polypols, which are nonlinear generalizations of polygons in the plane. The text is a collection of known results. It contains many examples and a list of exercises.