Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 114: Convex Geometry
Search by ISBN on Amazon: 9780486458038
If teaching from lecture notes:
Lecture(s) 
Sections 
Comments/Topics 
3 
Fundamental definitions: Affine sets, convex set, convex hull. Examples. 

3 
Caratheodory’s theorem, Radon’s theorem. 

3 
Helly’s theorem and applications. 

3 
Separating and supporting hyperplanes. Faces, extreme points. 

3 
Sets of constant width, diameter Borsuk’s problem. 

3 
Polyhedra and Polytopes. Examples and main operations (e.g. Projections, Schlegel Diagrams). 

3 
Graphs of polytopes, Euler’s formula. Coloring problems. 

3 
Duality and Polarity. 

3 
Convex bodies and Lattices. Minkowski’s first theorem, Blichfeldt’s theorem. 
If using Lay's book:
Lecture(s) 
Sections 
Comments/Topics 
2  Definition of convex sets, convex bodies, and convex polytopes. Examples of convex polytopes and nonpolytopes in dimensions 2 and 3. Examples of intuitive results and open problems, e.g., sphere packing. 

20  Euclidean and convex geometry in n dimensions. kdimensional faces of ndimensional polytopes. Volumes of parallelepipeds and simplices. Volume of the nsphere and multiple integrals for volumes. 

Wikipedia  Definition of a regular polytope. The list of regular npolytopes and enumeration of faces. Dihedral angles of regular polytopes and the necessary condition that the total angle of each ridge is less than 2*pi. 

2  Closures, convex hulls, and Caratheodory's theorem. 

3, 4  Existence of separating and supporting hyperplanes. 

5  Extreme points and the finitedimensional KreinMilman theorem.
The definition of kextreme points. The relation between kfacets and kextreme points. Examples of kextreme points in nonpolytopes. 

14  Arithmetic with sets and Minkowski sums. Statement of the BrunnMinkowski and RogersShephard inequalities. The isoperimetic inequality as a corollary of BrunnMinkowski. 

23  Polar duals of convex sets, convex bodies, and polytopes. The correspondence between the face poset of a polytope and its dual. 

Optional Advanced Topics  
The classification of convex polygons that tile the plane.  
The symmetrization proof of BrunnMinkowski inequality and the isoperimetric inequality.  
The topological proof of existence of regular polytopes. The definition of a Coxeter simplex.  
Sphere packings and Voronoi regions. Examples of sphere packings in higher dimensions.  
The largest ellipsoid in a convex body (the John ellipsoid). The definition of BanachMazur distance and the fact that it is bounded in each dimension. 