The Extremal Spheres Theorem

TL;DR

This paper examines the extremal neighboring spheres of polygons and polytopes, identifies gaps in a previous proof claiming a lower bound of 2d extremal spheres, and establishes a new lower bound of d+1, leaving the 2d case open.

Contribution

The paper critically analyzes Schatteman's 1990 proof, demonstrates its gaps, and improves the known lower bound of extremal neighboring spheres from 2 to d+1.

Findings

Identifies gaps in Schatteman's proof of the 2d extremal spheres conjecture.

Establishes a new lower bound of d+1 extremal neighboring spheres.

The existence of 2d extremal neighboring spheres remains an open problem.

Abstract

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in d-dimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres. In this paper, we show that there are certain gaps in Schatteman's proof, which is based on the Bruggesser-Mani shelling method. We show that using this method it is possible to prove that there are at least d+1 extremal neighboring spheres. However, the existence problem of 2d extremal neighboring spheres is still open.