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Illustrations

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Animations

Deformations permutahedron to associahedron to cube Construction monotone path polytope tetrahedron Construction pivot polytope tetrahedron
Sequence of deformations (from a permutahedron to an associahedron, then to a cube) Construction of the monotone path polytope of the tetrahedron Construction of the max-slope pivot rule polytope of the tetrahedron

Images

Random polytopes

Projection sphere to plane Plot of beta-distribution
The projection of the uniform distribution on the sphere to the plane gives a β-distribution Plot of β-distribution for β = -0.5, β = 1, β = 10
epsilon-cap and epsilon-floating body Region outside epsilon-floating body (rotationaly symmetric distribution)
Construction of an ε-cap and of the ε-floating body Region outside the ε-floating body for a rotationaly symmetric distribution
Visible region from a point First order difference operator
Definition of the region visible from a point, outside the ε-floating body Computation of the first difference operator

Monotone paths polytopes and pivot rule polytopes

Coherent monotone path Monotone path polytope as sum sections
Definition of the coherent monotone path associated to ω Construction of the monotone path polytope (of a tetrahedron) as a Minkowski sum of sections (of this tetrahedron)
Monotone path polytope via stereographic projection Monotone path polytope hypersimplex 4,2
Construction of (the normal fan of) the monotone path polytope of a tetrahedron via the stereographic projection of the normal of of this tetrahedron Construction of the monotone path polytope of the hypersimplex Δ(4, 2), first combinatorially, then as the convex hull of points associated to all monotone paths (interior points correspond to non-coherent paths)
Monotone path polytope hypersimplex 5,2 Pivot polytope as sum sections
Monotone path polytope of the hypersimplex Δ(5, 2) (not actual geometry) Construction of the pivot rule polytope (of a tetrahedron) as a Minkowski sum of sections (of this tetrahedron)
Pivot polytope of cube Pivot polytope of tetrahedron
Pivot rule polytope of the standard 3-dimensional cube Pivot rule polytope of the standard 3-dimensional simplex

Special polytopes

Lopsided cube Truncated lopsided cube
The lopsided cube of dimension 3 with its directed graph (its number of monotone paths per length is not unimodal) The truncated lopsided cube of dimension 3 (its number of monotone paths per length is not unimodal)
Simplcial polytope whose number of monotone paths per length is not unimodal Cyclic polytope dimension 3
A simplcial polytope whose number of monotone paths per length is not unimodal Cyclic polytope of dimesion 3 (its graph is not the complete graph)
Associahedron (dimension 2)
The 2-dimensional associahedron, labeled by multiple combinatorial families

Polytopes (general facts)

H- and V- polytopes Example bounded linear program
A (redundant) H-description and a (redundant) V-description of the same polytope Examples of (the bounded feasible domain of) a linear program

Submodular cone & Deformed permutahedra (a.k.a generalized permutahedra)

Permutahedra small dimensions Hexagon as a graphical zonotope
The first permutahedra, up to dimension 3 Hexagon: graphical zonotope of the 3-cycle
Submodular cone n = 3 Deformation cone rhombo-dodecahedron
The (intersection with an hyperplane of the) submodular cone for n = 3 (i.e. the deformation cone of the 3-dimensional permutahedron) The (intersection with an hyperplane of the) deformation cone of the rhombo-dodecahedron (i.e. of the graphical zonotope of the 4-cycle)
Rhombor-dodecahedron Rhombo-hexagonal-dodechadron
Rhombo-dodecahedron: graphical zonotope of the 4-cycle Rhombo-hexagonal-dodecahedron: graphical zonotope of the 4-cycle with a chord
Braid and sylvester fans
The braid fan and the sylvester fan of dimension 2

Combinatorics

Bijection triangulations vs non-crossing arboresecnces Flippable quandrangles vs flippable triplets
Bijection between triangulations of a polygon and non-crossing arborescences Flips of quandrangles correspond to flip of arcs