Unimodality and convexity of f-vectors of polytopes

TL;DR

This paper explores the unimodality and convexity properties of f-vectors of polytopes across different dimensions, reviewing known results, presenting new constructions, and proposing conjectures to advance understanding in polytope combinatorics.

Contribution

It provides an overview of current results on f-vector properties in higher dimensions, introduces a novel construction, and formulates a new conjecture.

Findings

f-vectors of 5-polytopes are unimodal

Presented a new polytope construction

Proposed a conjecture on higher-dimensional f-vectors

Abstract

We consider unimodality and related properties of f-vectors of polytopes in various dimensions. By a result of Kalai (1988), f-vectors of 5-polytopes are unimodal. In higher dimensions much less can be said; we give an overview on current results and present a potentially interesting construction as well as a conjecture arising from this.