Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids

TL;DR

This paper proves that in four dimensions, prismatoids cannot have a width greater than their dimension, providing insights into the structure of polytopes and implications for the Hirsch conjecture.

Contribution

It demonstrates that 4-dimensional prismatoids cannot have width larger than 4, extending previous results and using graph embeddings on a sphere to establish this.

Findings

4-dimensional prismatoids have width at most 4

Counterexamples to the Hirsch conjecture do not exist in dimension four

Graph embeddings on a sphere are key to the proof

Abstract

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The author recently showed in arXiv:1006.2814 that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases of $Q$.