Nonsimplicities and the perturbed wedge

TL;DR

This paper analyzes the technical details of Santos and Weibel's construction of a 20-dimensional counterexample to the Hirsch conjecture, focusing on the perturbed wedge operation and the evolution of nonsimplicities.

Contribution

It provides a detailed exploration of the primal construction process, starting from a non-simple 5-dimensional counterexample, and shows how the perturbed wedge operation leads to a simple polytope exceeding the Hirsch bound.

Findings

The initial polytope P5 contains nonsimple vertices.

Repeated perturbed wedge operations reduce nonsimplicity excess.

The final 20-dimensional polytope is simple and exceeds the Hirsch bound by 1.

Abstract

In 2010 Santos described the construction of a counterexample to the Hirsch conjecture, and in 2012 Santos and Weibel provided the coordinates for the 40 facets of a 20-dimensional counterexample. In this paper we explore technical details of the construction using Santos and Weibel's work as the motivating example. Santos presented the construction in the dual setting. Here we return to the primal setting, in which Santos' construction calls for repeated application of a perturbed wedge operation, a wedge over a facet followed by a perturbation of one or more other facets. We show that the starting point for the construction is a counterexample "P5" to the nonrevisiting conjecture in dimension 5. However, this polytope P5 is not a simple polytope; it contains two nonsimple vertices. As we repeatedly apply the perturbed wedge, the nonsimplicities grow in dimension while their excess is reduced. Finally in dimension 20, the resulting polytope is simple and its diameter exceeds the Hirsch bound by 1. These notes are a technical companion to the work of Santos and Weibel.