Determine the rays of the permutahedron’s deformation cone

Determine the rays of the deformation cone of the permutahedron, thereby resolving the longstanding open problem posed by Edmonds.

Background

The paper identifies the permutahedron as a zonotope whose deformation cone can nevertheless be highly complicated. In particular, although the permutahedron has a Minkowski-sum representation by line segments, its deformation cone has substantially larger dimension, and understanding its extremal rays remains a central unresolved problem in the study of deformation cones and Minkowski decompositions.

This problem is distinct from the paper’s results on 2-cubical zonotopes: the permutahedron is not 2-cubical because some of its 2-faces are hexagons, so the upper-bound techniques developed in the paper do not resolve Edmonds’s question.