Polytopes

Lectures on Polytopes

1. 1 Polytopes, Polyhedra, and Cones
• Definitions: H-polytope, V-Polyhedron, V-Polytope, Cone, Conical Hull
• Minkowski–Weyl: \(\mathcal{V}\)-polyhedron \( \implies \mathcal{H}\)-polyhedron [using Fourier-Motzkin Elimination]
• Minkowski–Weyl: \(\mathcal{H}\)-polyhedron \( \implies \mathcal{V}\)-polyhedron [TODO]
• Proposition 1.7: Farkas Lemma (I) (Incomplete Notes) [TODO]
• Proposition 1.8: Farkas Lemma (II)
• Proposition 1.9: Farkas Lemma (III)
2. 2 Faces of Polytopes
• Definitions: Faces of Polytopes
• Proposition 2.2: Every polytope is the convex hull of its vertices
• Proposition 2.3(i): The face \(F\) is a polytope, with \(\operatorname{vert}(F)=F\cap V\).
• Proposition 2.3(ii): Every intersection of faces of \(P\) is a face of \(P\).
• Proposition 2.3(iii): The faces of \(F\) are exactly the faces of \(P\) that are contained in \(F\).
• Definitions: Poset, Graded Poset, Lattice,... (+ Example)
• Theorem 2.7(i): For every polytope \(P\), the face poset \(L(P)\) is a graded lattice of length \(\dim(P)+1\), with rank function \(r(F)=\dim(F)+1.\)