Multitriangulations, pseudotriangulations and some problems of realization of polytopes

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.

💡 Research Summary

This dissertation investigates two seemingly distinct topics in discrete geometry—multitriangulations and the polytopal realizations of Cartesian products—through the unifying lens of “realizing a combinatorial structure as a polytope.”
The first part introduces the notion of a k‑triangulation (or multitriangulation) of a convex n‑gon: a maximal set of chords with the property that no k + 1 of them mutually cross. When k = 1 we recover ordinary triangulations; for larger k the elementary building blocks are no longer triangles but “stars,” each consisting of k + 1 chords meeting at a common interior point without crossing one another. By focusing on stars the author obtains a recursive decomposition of any k‑triangulation and defines a natural flip operation: replace the unique chord shared by two adjacent stars with the complementary chord, thereby producing a new star. The collection of all k‑triangulations together with these flips forms a connected flip graph.
A central insight is the duality between this flip graph and arrangements of pseudolines with contact points. One can view the n‑gon’s sides as a fixed support and embed each star as a contact point of a pseudoline arrangement that covers the support. Flipping a star corresponds exactly to moving a contact point in the pseudoline picture, which shows that the flip graph is isomorphic to the graph of admissible moves in the pseudoline arrangement. This duality mirrors the classical correspondence between ordinary triangulations and planar graphs, but it extends it to higher‑order crossing constraints.
The author then asks whether the flip graph of k‑triangulations can be realized as the 1‑skeleton of a convex polytope, i.e., whether there exists a “k‑associahedron.” For k = 1 the answer is affirmative (the classical associahedron). For k > 1 only partial results are known: explicit realizations exist for small n and for k = 2, but a general construction remains elusive. The star‑pseudoline framework suggests a possible coordinate construction, yet the required dimension grows rapidly and the resulting inequalities are highly non‑linear, preventing a clean, uniform polytope description. Consequently, the existence of a universal k‑associahedron is posed as an open problem.
The second part shifts focus to the polytopality of Cartesian products. The question is twofold: (i) when does the Cartesian product of two graphs admit a polyhedral realization whose 1‑skeleton is exactly the product graph, and (ii) what is the minimal dimension of a polytope whose k‑skeleton coincides with that of a product of simplices. For complete graphs K_m and K_n, the product K_m□K_n can be realized as the 1‑skeleton of a polytope of dimension at least m + n − 2, a bound that follows from simple counting of facets and vertices. The author revisits this result, providing constructive embeddings based on Minkowski sums and Cayley embeddings.
When the factors are simplices Δ^p and Δ^q, the product Δ^p□Δ^q is a (p + q)-dimensional polytope whose faces are products of faces of the factors. The thesis investigates how much one can lower the ambient dimension while preserving the k‑skeleton. Using the Cayley trick, the author shows that for small k (relative to p and q) the product can be embedded in dimension p + q, but as k grows one needs additional dimensions; a lower bound of p + q + k − 1 is derived for the minimal possible dimension. These results bridge the gap between combinatorial descriptions of product complexes and their geometric realizations.
Finally, the dissertation lists several open problems: (1) the existence (or non‑existence) of a universal k‑associahedron for all k and n; (2) algorithmic methods to compute explicit coordinates for the star‑pseudoline duality; (3) tight dimension bounds for polyhedral realizations of arbitrary graph products; (4) structural properties of “skeleton polytopes” that realize only a prescribed k‑skeleton while minimizing dimension. By connecting multitriangulations, pseudoline arrangements, and product polytopes, the work opens new avenues for research at the intersection of combinatorial geometry, polytope theory, and computational geometry.