  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.
