Depth in arrangements: Dehn-Sommerville-Euler relations with applications (Q6645901)

In the paper under review, the authors use topological methods to understand combinatorial properties of certain arrangements. Here the authors focus on the depth of a cell in an arrangement of \(n\) (non-vertical) great-spheres in \(\mathbb{S}^{d}\) which is the number of great-spheres that pass above the cell. The first result is devoted to Euler-type relations, which imply extensions of the classic Dehn-Sommerville relations for convex polytopes to sublevel sets of the depth function. Then the authors use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.