Discrete Geometry II
Summer Semester 2015
news
times
syllabus
literature
homeworks
what happened so far
Prof. Raman Sanyal, Arnimallee 2, room 105
(office hours: TBA)
Dr. Arnau Padrol , Arnimallee 2, room 103 (office hours:
TBA)
news
All updates will posted on the
mailing list.
Please make sure you subscribed!
 the wiki is on
 new lecture times!
 class starts Tuesday, April 14
 please sign up on the mailing
list
times
lecture 
tuesday 
1012 
Arnimallee 6 SR 007/008 

wednesday 
1416 
Arnimallee 6 SR 007/008 

recitation 
thursday 
1012 
Arnimallee 3 SR 119 

syllabus / prerequisites / formalities
This is the second in a series of three courses on
discrete geometry. The aim of the course is a skillful
handling of discrete geometric structures with an emphasis
on convex geometric properties. In the course we will
develop central themes in discrete and convex geometry
including proof techniques and applications to other areas
in mathematics.
The material will be a selection of the following topics:
Basic structures in discrete and convex geometry
 convexity and separation theorems
 convex bodies and polytopes/polyhedra
 faces and boundary structures
 polarity
 approximation by polytopes
 subdivisions and triangulations (including
Delaunay and Voronoii)
Notions and tools
 face numbers and invariants
 volumes and mixed volumes
 volume computations and estimates
 LöwnerJohn ellipsoids and roundness
 valuations
Geometric inequalities
 face numbers and invariants
 BrunnMinkowski and AlexsandrovFenchel inequality
 isoperimetric inequalities
 measure concentration and phenomena in highdimensions
 Minkowski's representation theorem for polytopes
 face numbers and invariants
Geometry of numbers
 lattices
 Minkowski's (first) theorem
 successive minima
 lattice points in convex bodies and Ehrhart's theorem
 EhrhartMacdonald reciprocity
Sphere packings
 lattice packings and coverings
 Theorem of MinkowskiHlawka
 analytic methods
The target audience are students with an interest in
discrete mathematics and (convex) geometry. The course is
a good entry point for a specialization in discrete
geometry. The topics addressed in the course supplement
and deepen the understanding for discretegeometric
structures appearing in differential geometry, topology,
combinatorics, and algebraic geometry.
prerequisites
Solid background in linear algebra and some analysis. Basic knowledge
and experience with polytopes and/or
convexity will be helpful.
Formalities
To successfully pass the course, you need to...
 ...actively participate in the recitations,
 ...get at least 60 percent of the total number of homework
points (see homeworks),
 ...present at least one of the homework problems in the
recitation, and
 ...pass the exam at the end of the semester (details will be
discussed in the first week).
literature
A good part of the mentioned books is in the
Semesterapparat at
the math library.
 Rolf Schneider,
Convex bodies: the BrunnMinkowski theory, Cambridge
University Press
 Alexander Barvinok,
A Course in Convexity, GSM 54, American Mathematical
Society
 Peter Gruber
Convex and discrete Geometry, Springer Grundlehren 336
 Günter Ewald,
Combinatorial Convexity and Algebraic Geometry , Springer
GTM 168
 Jirka Matousek,
Lectures on Discrete Geometry, Springer GTM 212
 Igor Pak,
Lectures on Discrete and Polyhedral Geometry
, preprint
 Günter Ziegler,
Lectures on Polytopes, Springer GTM 152
 T. Bonnesen, W. Fenchel,
Theorie der konvexen Körper, American Mathematical
Society
homeworks
There will be one homework sheet per week (posted here).
You have to hand them in Wednesday
before the
lecture. More information will come soon.
Hand in the solutions in
pairs.
Let us know if you couldn't find a partner. Try to solve all
the problems but
mark two solutions. Only these
will be
graded. (The supplementary problems do not count; they are
for your enjoyment.)
You can earn 20 points on every homework sheet
(10 per exercise). There are marked problems that are
mandatory.
You can get extra credit by solving the bonus problems.
State who wrote up the solution. Everybody has to
write
up at least 25 percent of the solutions. Everybody has to
present
at least one problem in the recitation session.
Everybody has to
write down the solution of one problem
in the
wiki, it will be graded as an additional homework sheet.
You can get extra credit by adding and editing entries of the wiki.
You can add/edit solutions for some homework exercises in the wiki:
http://www.mathseminar.org/DiscreteGeometryII.
what happened so far
You can find LaTeXed lecture notes
here.
Disclaimer: they are not necessarily complete, they are not
necessarily correct. Don't rely on these notes  take your own!
If you find errors or shortcomings, please let me know.
May 12/13

flags of faces; barycentric subdivision; pyramids and
volumes;
general formula for volume; Minkowski's theorem
and the easy half; beneathbeyond and placing
triangulations  a practical algorithm

May 5/6

distance function of a convex body; membership/separation
oracles; outer parallel bodies and Hausdorff distance;
Relation to support functions; approximations of convex
bodies by polytopes;
Volume of convex bodies I: Volume as Jordan measure;
polyboxes and Jordan measure; Volume from scratch:
dissections and Hilbert's 3rd problem; Volume of a simplex
and extensions to all polytopes;

April 28/29

polarity; properties; MinkowskiWeyl for polytopes; faces,
facets; conjugate faces
finitely generated and polyhedral cones; polars of cones;
Farkas lemma; digression: linear programming, weak and
strong duality; support functions; Theorem: support
functions are exactly the sublinear (positively
homogeneous and subadditive) functions; the convex cone of
convex bodies

April 21/22

extreme points and Minkowski's theorem: A convex body is
the convex hull of its extreme points; nearest point maps
characterize closed convex sets; separating
hyperplanes and a separation theorem; supporting
hyperplanes and the support function; the monoid of convex
bodies;
proof of Minkowski's theorem; faces and exposed faces;
face lattice; faces vs. exposed faces; faces of polytopes
are exposed

April 14/15

5 minutes to fall in love with discrete convex geometry;
linear/affine subspaces, linear/affine hulls; segments and
convex sets; Examples: unit balls and norms, convex cone
of positive semidefinite matrices
Examples (cont'd): positive polynomials, sumsofsquares,
epigraphs and convex functions; convex hulls and
polytopes; convex combinations; Caratheodory's theorem;
(relative) interior and boundary; affine hull and
dimension; convex hull of compact set is compact
