Germain Poullot
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# Illustrations

You can copy any of my images in order to use it as an illustration for your presentations, pre-print, courses, exercises, etc. However, before submitting a paper for publication, if you are using one of my images, please email me to ask permission (I will almost surely say "yes").

When reproducing one of these images, please cite my name (G. Poullot), and, if time and space allows use, cite the year, and/or the article the image is extracted from.

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## Animations

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

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
Construction of an ε-cap and of the ε-floating body | Region outside the ε-floating body for a rotationaly symmetric distribution
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

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)
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 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 rule polytope of the standard 3-dimensional cube | Pivot rule polytope of the standard 3-dimensional simplex

### Special polytopes

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)
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)
The 2-dimensional associahedron, labeled by multiple combinatorial families | Graphical zonotope of the 4 cycle
Rhombo-dodecahedron: graphical zonotope of the 4-cycle | Rhombo-hexagonal-dodecahedron: graphical zonotope of the 4-cycle with a chord
Standard simplex of dimension 3 | Pyramid over a square
Octahedron | Strawberry
Persimmon

### Polytopes (general facts)

A (redundant) H-description and a (redundant) V-description of the same polytope | Examples of (the bounded feasible domain of) a linear program
Wedge over a polytope in direction e_(n+1) - e_i

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

The first permutahedra, up to dimension 3 | Hexagon: graphical zonotope of the 3-cycle
The braid fan and the sylvester fan of dimension 2 | The construction of the GP-sum
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)
The (intersection with an hyperplane of the) submodular cone for n = 3, as GP-sums | Partition of the (intersection with an hyperplane of the) submodular cone for n = 3 via generic GP-sums
Part of the face lattice of the submodular cone above the (deformation cone of the) cube | Partition of the face lattice of the (deformation cone of the) a generic GP-sum into (deformation cones of) generic GP-sums

### Minkowski sums and indecomposability

Regular written as Minkowski sums of indecomposable polytopes | Proof of the indecomposability of an octahedron
Proof of the indecomposability of the Strawberry | Proof of the indecomposability of the cuboctahedron
Proof of the indecomposability of the Persimmon | Kallay's example of combinatorially equivalent polytopes: one decomposable, one indecomposable

### Combinatorics

Bijection between triangulations of a polygon and non-crossing arborescences | Flips of quandrangles correspond to flip of arcs