Generalizations of cyclic polytopes

TL;DR

This paper reviews generalizations of cyclic polytopes, highlighting their properties, applications, and potential future directions in mathematical research.

Contribution

It provides an overview of various generalizations of cyclic polytopes, emphasizing their explicit structures and applications across different mathematical fields.

Findings

Generalizations preserve key properties of cyclic polytopes.

New classes of polytopes with explicit facet structures are identified.

Applications span multiple branches of mathematics.

Abstract

Cyclic polytopes have been studied since at least the early last century by Caratheodory and others.A generalization is a construction of a class of polytopes such that the polytopes have some of their properties.The best known example is the class of neighbourly polytopes. Cyclic polytopes have explicit facet structures, important properties and applications in different branches of mathematics. In the past few decades, generalizations of their combinatorial properties have yielded new classes of polytopes that also have explicit facet structures and useful applications. We present an overview of these generalizations along with some applications of the resultant polytopes, and some possible approaches to other generalizations.