Geometrical realisations of the simple permutoassociahedron by Minkowski sums

This paper introduces a family of n-polytopes, PAn,c which is a geometrical realisation of simple permutoassociahedra. It has significant importance serving as a topological proof of Mac Lane's coherence. Polytopes in this family are defined as Minkowski sums of certain polytopes such that every summand produces exactly one truncation of the permutohedron, i.e. yields to the appropriate facet of the resulting sum. Additionally, it leads to the correlation between Minkowski sums and truncations, which gives a general procedure for similar geometrical realisation of a wider class of polytopes.