Conductive rings of nonpolytopal fans

Abstract

Although there is a lot of polytopality criteria for fans, it is generally not clear how to visualize the reason why a fan is (non)polytopal. The paper describes a geometric structure called a conductive ring that is easy to find visually in fans. Found in a fan, a conductive ring means that the fan is nonpolytopal. The paper shows that conductive rings provide a very simple and useful tool for analysing, modifying, and creating fans. In particular, constructing a nonpolytopal fan with a conductive ring, the paper discovers a paradox concerning intermediate value properties of face polytopes: in a bundle of very similar conditions the properties either reveal themselves or not depending on small details.