A tale of centrally symmetric polytopes and spheres

TL;DR

This survey reviews recent progress and open problems in the combinatorial and geometric properties of centrally symmetric polytopes and spheres, focusing on face numbers and related theorems.

Contribution

It compiles and discusses recent advances, open questions, and conjectures in the study of face numbers of centrally symmetric polytopes and spheres.

Findings

Overview of neighborliness and upper bound theorems for centrally symmetric polytopes

Discussion of the generalized lower bound theorem for these structures

Presentation of the lower bound conjecture for centrally symmetric spheres and manifolds

Abstract

This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.