The excess degree of a polytope

TL;DR

This paper introduces the excess degree of a polytope as a measure of deviation from simplicity, explores its possible values, and applies it to classify certain polytopes and their decomposability properties.

Contribution

It defines the excess degree for polytopes, analyzes its possible values, and applies this concept to classify polytopes with small excess, decomposable polytopes with specific vertices, and pairs of vertices and edges in 5-polytopes.

Findings

Excess degree does not take all natural numbers; smallest values are 0 and d-2.

Polytopes with small excess are either decomposable or pyramidal, with indecomposable duals.

Complete characterization of decomposable d-polytopes with 2d+1 vertices.

Abstract

We define the excess degree $\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple. It turns out that the excess degree of a $d$-polytope does not take every natural number: the smallest possible values are $0$ and $d-2$, and the value $d-1$ only occurs when $d=3$ or 5. On the other hand, for fixed $d$, the number of values not taken by the excess degree is finite if $d$ is odd, and the number of even values not taken by the excess degree is finite if $d$ is even. The excess degree is then applied in three different settings. It is used to show that polytopes with small excess (i.e. $\xi(P)<d$) have a very particular structure: provided $d\ne5$, either there is a unique nonsimple vertex, or every nonsimple vertex has degree $d+1$. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Secondly, we characterise completely the decomposable $d$-polytopes with $2d+1$ vertices (up to combinatorial equivalence). And thirdly all pairs $(f_0,f_1)$, for which there exists a 5-polytope with $f_0$ vertices and $f_1$ edges, are determined.