Vertex generated polytopes

TL;DR

This paper introduces the class of vertex generated polytopes, explores their properties, and demonstrates their density in two dimensions, along with applications to polytope approximation and covering properties.

Contribution

It defines vertex generated polytopes, proves their inclusion of zonotopes, density in 2D, and establishes approximation and covering results related to this class.

Findings

All zonotopes are vertex generated.

Vertex generated polytopes are dense in 2D.

Any polytope can be summed with a zonotope to become vertex generated.

Abstract

In this paper we define and investigate a class of polytopes which we call "vertex generated" consisting of polytopes which are the average of their $0$ and $n$ dimensional faces. We show many results regarding this class, among them: that the class contains all zonotopes, that it is dense in dimension $n=2$, that any polytope can be summed with a zonotope so that the sum is in this class, and that a strong form of the celebrated "Maurey Lemma" holds for polytopes in this class. We introduce for every polytope a parameter which measures how far it is from being vertex-generated, and show that when this parameter is small, strong covering properties hold.