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

## TL;DR

This paper explores mathematical properties of cell arrangements in high-dimensional spheres, extending known geometric relations to new contexts.

## Contribution

The paper introduces new Euler-type relations for depth in arrangements of great-spheres, extending Dehn–Sommerville relations to sublevel sets.

## Key findings

- Euler-type relations are proven for depth in arrangements of great-spheres in spheres.
- Dehn–Sommerville relations are extended to sublevel sets of the depth function.
- Face counts for neighborly polytopes are generalized to cell counts in neighborly arrangements.

## Abstract

The depth of a cell in an arrangement of n (non-vertical) great-spheres in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^d$$\end{document}Sd 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.