Convex Polytopes: Extremal Constructions and f-Vector Shapes

TL;DR

This paper explores the geometric properties of convex polytopes, focusing on f-vector shapes and extremal constructions, including proofs, high-dimensional analysis, and recent constructions with high fatness.

Contribution

It provides new insights into the shapes of f-vectors, detailed proofs of key theorems, and introduces recent constructions achieving high fatness in 4-polytopes.

Findings

Complete proof of the Koebe--Andreev--Thurston theorem for 3-polytopes

Analysis of f-vector shapes in high-dimensional polytopes

Construction of 4-polytopes with fatness approaching 9

Abstract

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of the cone of f-vectors for 4-polytopes, and thus identifies the existence/construction of 4-polytopes of high ``fatness'' as a key problem. In this direction, the last lecture presents a very recent construction of ``projected products of polygons,'' whose fatness reaches 9-\eps.