Generalized blade arrangements at varying vertices

Determine which collections of general blades placed at different vertices induce polymatroidal blade arrangements, thereby combining variation in the ordered set partitions defining the blades with variation in their translations.

Background

The paper characterizes compatibility for collections of alcoved simplices, and uses this characterization to obtain polymatroidal blade arrangements when the blades are based at a common point. It also recalls a complete criterion for arrangements of the standard blade placed at different hypersimplex vertices. The authors identify the unresolved generalization in which both the cyclic orders or ordered set partitions and the translation vertices vary simultaneously.