Depth in arrangements: Dehn–Sommerville–Euler relations with applications

The depth of a cell in an arrangement of n (non-vertical) great-spheres in 𝕊^d is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we 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.