# Geometrical Realisations of the Simple Permutoassociahedron by Minkowski sums

## TL;DR

This paper presents a new geometric construction of simple permutoassociahedra using Minkowski sums, providing a topological proof of Mac Lane's coherence and a general method for constructing permutonestohedra.

## Contribution

It introduces a family of polytopes formed by Minkowski sums that realize permutoassociahedra and links Minkowski sums to permutohedron truncations for broader applications.

## Key findings

- Established a Minkowski sum-based geometric realization of permutoassociahedra.
- Connected Minkowski sums with permutohedron truncations.
- Provided a general Minkowski-construction procedure for permutonestohedra.

## Abstract

This paper offers a geometrical realisation of simple permutoassociahedra, which has significant importance serving as a topological proof of Mac Lane's coherence. We introduce a family of $n$-polytopes, $PA_{n,c}$, obtained by Minkowski sums such that each summand yields to the appropriate facet of the resulting sum. Additionally, it leads to the correlation between Minkowski sums and truncations of permutohedron, which implicitly gives a general procedure for geometrical Minkowski-construction of any permutonestohedron.