Realizing the associahedron: Mysteries and questions

TL;DR

This paper explores open problems and mysteries related to the convex polytope realizations of associahedra, including special cases, the space of all realizations, and extensions to multiassociahedra.

Contribution

It identifies and discusses three key open problems in the realization of associahedra and multiassociahedra, highlighting areas for future research.

Findings

Identified open problems in associahedron realizations

Discussed special realizations with vertices on a sphere

Explored the space of all possible realizations

Abstract

There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe three -- concerning special realizations with the vertices on a sphere, the space of all possible realizations, and possible realizations of the multiassociahedron.