Unimodality of f-vectors of cyclic polytopes

TL;DR

This paper proves that the f-vectors of cyclic polytopes are unimodal, highlighting a fundamental property that contrasts with their use in constructing non-unimodal convex polytopes.

Contribution

It establishes the unimodality of f-vectors of cyclic polytopes, a property less known compared to their role in the Upper Bound Theorem.

Findings

F-vectors of cyclic polytopes are unimodal.

Cyclic polytopes have a special shape of f-vectors.

Unimodality contrasts with their application in non-unimodal constructions.

Abstract

Cyclic polytopes are generally known for being involved in the Upper Bound Theorem, but they have another extremal property which is less well known. Namely, the special shape of their f-vectors makes them applicable to certain constructions to present non-unimodal convex polytopes. Nevertheless, the f-vectors of cyclic polytopes themselves are unimodal.