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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/217199
Geometric realizations using regular subdivisions: Construction of many polytopes, sweep polytopes, s-permutahedra
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[eng] This thesis concerns three problems of geometric realizations of combinatorial structures via polytopes and polyhedral subdivisions. A polytope is the convex hull of a finite set of points in a Euclidean space Rd. It is endowed with a combinatorial structure coming from its faces. A subdivision is a collection of polytopes whose faces intersect properly and such that their union is convex. It is regular if it can be obtained by taking the lower faces of a lifting of its vertices in one dimension higher. We first present a new geometric construction of many combinatorially different polytopes of fixed dimension and number of vertices. This construction relies on showing that certain polytopes admit many regular triangulations. It allows us to improve the best known lower bound on the number of combinatorial types of polytopes. We then study the projections of permutahedra, that we call sweep polytopes because they model the possible orderings of a fixed point configuration by hyperplanes that sweep the space in a constant direction. We also introduce and study a combinatorial abstraction of these structures: the sweep oriented matroids, that generalize Goodman and Pollack’s theory of allowable sequences to dimensions higher than 2. Finally, we provide geometric realizations of the s-weak order, a combinatorial structure that generalizes the weak order on permutations, parameterized by a vector s ∈ (Z>0)n. In particular, we answer Ceballos and Pons’s conjecture that the s-weak order can be realized as the edge-graph of a polytopal complex that is moreover a subdivision of a permutahedron.
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PHILIPPE, Eva. Geometric realizations using regular subdivisions: Construction of many polytopes, sweep polytopes, s-permutahedra. [consulta: 30 de novembre de 2025]. [Disponible a: https://hdl.handle.net/2445/217199]