17 citations · 73 across the 26 of their papers we have counts for
33 papers
Reconstruction of slices of cubes
Marie-Charlotte Brandenburg, Juliana Curtis, Chiara Giardino +2
We prove that the combinatorial type of a slice of the cube by a hyperplane is completely determined by its graph, within the class of cube slices. We also show that such a result…
Indecomposability of 0/1-polytopes
Akihiro Higashitani, Arnau Padrol, Raman Sanyal
We prove that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspace…
The graph of implicit edge dependencies for indecomposability and beyond
Arnau Padrol, Germain Poullot
A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly st…
Framed Polytopes and Higher Categories
Guillaume Laplante-Anfossi, Anibal M. Medina-Mardones, Arnau Padrol
In the early 1990s, Kapranov and Voevodsky proposed a geometric method for constructing higher-categorical pasting diagrams from generically framed convex polytopes. This work revi…
Many rays of the submodular cone
Georg Loho, Arnau Padrol, Germain Poullot
The study of the cone of submodular functions goes back to Jack Edmonds' seminal 1970 paper, which already highlighted the difficulty of characterizing its extreme rays. Since then…
Facial nested complexes and acyclonestohedra
Chiara Mantovani, Arnau Padrol, Vincent Pilaud
We study nested complexes of building sets on the Las Vergnas face lattices of oriented matroids. Such a nested complex is the face lattice of an oriented matroid, obtained by iter…
An asymptotic rigidity property from the realizability of chirotope extensions
Xavier Goaoc, Arnau Padrol
Let be a finite full-dimensional point configuration in . We show that if a point configuration has the property that all finite chirotopes realizable by addi…
Many regular triangulations and many polytopes
Arnau Padrol, Eva Philippe, Francisco Santos
We show that for fixed and growing to infinity there are at least different labeled combinatorial types of -polytopes with vertices. This is…
Deformed graphical zonotopes
Arnau Padrol, Vincent Pilaud, Germain Poullot
We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in r…
Deformation cones of graph associahedra and nestohedra
Arnau Padrol, Vincent Pilaud, Germain Poullot
We give the facet description of the deformation cones of graph associahedra and nestohedra, generalizing the classical parametrization of the family of deformed permutahedra by th…
An effective solution to convex -body -representability
Federico Castillo, Jean-Philippe Labbé, Julia Liebert +3
From a geometric point of view, Pauli's exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of -fermion and -fermion density matrices…
Sweeps, polytopes, oriented matroids, and allowable graphs of permutations
Arnau Padrol, Eva Philippe
A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman…
Shard polytopes
Arnau Padrol, Vincent Pilaud, Julian Ritter
For any lattice congruence of the weak order on permutations, N. Reading proved that gluing together the cones of the braid fan that belong to the same congruence class defines a c…
The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem
Leonardo Martínez-Sandoval, Arnau Padrol
The convex dimension of a -uniform hypergraph is the smallest dimension for which there is an injective mapping of its vertices into such that the set of -…
Associahedra for finite type cluster algebras and minimal relations between -vectors
Arnau Padrol, Yann Palu, Vincent Pilaud +1
We show that the mesh mutations are the minimal relations among the -vectors with respect to any initial seed in any finite type cluster algebra. We then use this a…
The -Tamari lattice via -trees, -bracket vectors, and subword complexes
Cesar Ceballos, Arnau Padrol, Camilo Sarmiento
We give new interpretations of the -Tamari lattice of Préville-Ratelle and Viennot. First, we describe it as a rotation lattice of -trees, which uncovers the relation with kn…
Geometry of -Tamari lattices in types and
Cesar Ceballos, Arnau Padrol, Camilo Sarmiento
In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of -Tamari latt…
Colorful simplicial depth, Minkowski sums, and generalized Gale transforms
Karim Adiprasito, Philip Brinkmann, Arnau Padrol +3
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained…
Extension complexity of polytopes with few vertices or facets
Arnau Padrol
We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of -polytopes with at most vertices accor…
Extension complexity and realization spaces of hypersimplices
Francesco Grande, Arnau Padrol, Raman Sanyal
The (n,k)-hypersimplex is the convex hull of all 0/1-vectors of length n with coordinate sum k. We explicitly determine the extension complexity of all hypersimplices as well as of…
Six topics on inscribable polytopes
Arnau Padrol, Günter M. Ziegler
Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six…
Scribability problems for polytopes
Hao Chen, Arnau Padrol
In this paper we study various scribability problems for polytopes. We begin with the classical -scribability problem proposed by Steiner and generalized by Schulte, which asks…
Polytopes with few vertices and few facets
Arnau Padrol
In this note we prove that the number of combinatorial types of -polytopes with vertices and facets is bounded by a constant independent of .
Enumerating neighborly polytopes and oriented matroids
Hiroyuki Miyata, Arnau Padrol
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form…