Multitriangulations, pseudotriangulations and some problems of realization of polytopes

TL;DR

This thesis investigates the combinatorial and geometric properties of multitriangulations and explores the polytopal realizations of products, addressing open problems in discrete geometry and polytope theory.

Contribution

It introduces a duality-based framework for multitriangulations and examines conditions for their polytopal realizations, as well as studying the minimal dimensions for polytopal realizations of product graphs.

Findings

Duality interpretation of multitriangulations as pseudoline arrangements.

Open problems on the polytopal realization of flip graphs.

Results on minimal dimensions for polytopal realizations of product graphs.

Abstract

This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitriangulations based on their stars, which play the same role as triangles of triangulations. This study leads to interpret multitriangulations by duality as pseudoline arrangements with contact points covering a given support. We exploit finally these results to discuss some open problems on multitriangulations, in particular the question of the polytopal realization of their flip graphs. We study secondly the polytopality of Cartesian products. We investigate the existence of polytopal realizations of cartesian products of graphs, and we study the minimal dimension that can have a polytope whose k-skeleton is that of a product of simplices.