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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Federico Ardila

Lecture 1: Course is a combinatorial focus on convex polytopes and hyperplane arrangements.

• Euler's theorem: 1 - v + e - f + 1 = 0
• Steinitz's theorem: A 3-polytope exists iff 1 - v + e - f + 1 = 0, v <= 2f - 4, f <= 2v - 4.
• Regular polytopes.

Lecture 2: Definition of polytope P as convex hull of vertices. Sum of lambda x vertex where lambdas are nonnnegative and sum to 1.

Lecture 3: Intersections and products of polytopes are also polytopes.

• Main theorem of polytopes: Polytopes = convex hulls of finitely many points = bounded intersections of halfspaces.

Lecture 4: V-polyhedron intersected with affine plane is a V-polyhedron.

Lecture 5 and 6: Farkas' lemma versions 1 to 4.

Lecture 7: Faces of polytopes. Face of P in direction c is the set of all x in P where c-x is maximal.

• Affine spaces.
• Dim(face) = Dim(Aff(face))
• f-vector gives the number of faces in each dimension
• f-poly the generated function where the lowest coefficient gives the number of vertices and the highest coefficient gives the volume

Lecture 8: Construction of faces

• Building polytopes: pyramids (adding the center).
• Vertex figures (converting the vertices to faces).
• Face lattice

Lecture 9: Face lattice

• P and Q are combinatorially isomorphic if their face lattices are isomorphic.
• Polar (Dual) polytopes. Dual = c in dual space where c x <=1 for all x in P.
• If 0 is in P, then P equals its dual's dual.

Lecture 10:

• Dual faces.
• The face lattices of P and its dual are opposites.
• Simple and simplicial polytopes.
• P is simple iff its dual is simplicial.

Lecture 11: The cyclic polytope.

Lecture 12: Graphs of polytopes

Lecture 13: How good is linear programming?

• Hirsch conjecture is false.

Lecture 14: Balinski's theorem: P is a d-polytope implies G(P) is d-connected.

Lecture 15: If P is simple, then G(P) determines P combinatorially.

Lecture 16: Complexes, subdivisions, triangulations.

• Every P has a triangulation.

Lecture 17: Triangulation of d-cross-polytope.

Lecture 18: Counting lattice points in polytopes.

Lecture 19: Partition functions.

Lecture 20: Generating functions for cones.

#### Polytopes

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