It was perfect, it was rounded, symmetrical, complete, colossal!
No seminar on 21.1.2014.
No seminar on 17.12.2013.
No seminar on 10.12.2013.
No seminar on 12.11.2013.
First (organizational) meeting is Tuesday, October 15, at 14:00 in the Seminar room in Arnimallee 2. Topics will be assigned there.
There will be ONLY one meeting per week (currently Tuesdays 1416)!
Runde Sachen are very appealing objects and the notion of roundness is very figurative. Mathematically, however, this is very difficult to grasp and in fact there are many different notions of roundness. The goal is to explore some of the central versions of `roundness' from a convexgeometric and from a discretegeometric perspective. In particular, we will see the interdependence of geometry and combinatorics. This is the central theme of the project A3 in the new Sonderforschungsbereich 'Discretization in Geometry and Dynamics'. The seminar builds a bridge between fundamental and classical results and ongoing research. Some topics are more challenging than others. A solid background in discrete geometry (e.g. Discrete Geometry I and/or II) is advised but no specialized knowledge is required.
A selection of topics for the seminar is this. Löwner–John roundness 

When  Who  What  References 

22.10.2013  Thomas Stollin  Löwner–John ellipsoids.A convex body is considered to be round if it is `close' to a ball. A classical result of Löwner and John states that, up to an affine transformation, every ddimensional convex body can be sandwiched between the unit ball and the dth dilate of the unit ball. In particular, the circumscribed ball is uniquely determined by unit `points of contact'. For centrallysymmetric convex bodies, the dilation factor for the LöwnerJohn ellipsoid can be decreased to sqrt(d). 

29.10.2013  Olaf Parczyk / Martin Skrodzki  The ellipsoid method.The ellipsoid method is a polynomialtime algorithm for solving linear programms. It proceeds by iteratively approximating a convex polyhedron by smaller and smaller ellipsoids and it is based on the existence of LöwnerJohn ellipsoids. Although this algorithm is not used in practice, it is an important theoretical tool. 

5.11.2013  Olaf Parczyk / Martin Skrodzki  The ellipsoid method with a simplex.One can mimic the strategy of the ellipsoid method with a simplex, obtaining a new algorithm for linear programming. 

Centrallysymmetric convex bodies 

When  Who  What  References 
19.11.2013  Christina Schulz  Neighborliness of centrallysymmetric polytopes.Neighborliness for centrallysymmetric polytopes measures how similar it is to the facial structure of a crosspolytope. LöwnerJohnround centrallysymmetric polytopes cannot be very neighborly. Indeed, if k(d,n) denotes the largest integer k such that there exists a kneighborly centrally symmetric dpolytope with 2(n+d) vertices, then k(d,n)= Θ(d/(1 + log((d + n)/d))). 

14.1.2014  Francesco Grande  Random centrally symmetric polytopes.There are several ways to consider a random centrally symmetric polytope. One of them is a random projection of an ndimensional crosspolytope onto R^d. When n is large and proportional to d, then these random centrally symmetric polytopes present some neighborliness with probability close to one. 

Round vs Pointy 

When  Who  What  References 
26.11.2013  Michael Brückner  Mahler volume.The Mahler conjecture is a long standing open problem in convex geometry. The Mahler volume M(B) of a centrally symmetric convex body B is M(B)=vol(B)vol(B*), where B* is the polar of B. Then the conjecture states that the minimum possible Mahler volume is attained by cubes and crosspolytopes. 

3.12.2013  Adem Güngör  Kalai's 3^d conjecture.In 1989 Gil Kalai conjectured that every ddimensional centrally symmetric polytope has at least 3^d nonempty faces. Although it is known that the conjecture is true for d≤4 and for simplicial polytopes (thanks to Bárány and Lovász's result) it remains open for the general case. 

7.1.2013  MarieSophie Litz  Simple centrally symmetric polytopes.★Using topological methods, one can prove that any simple centrally symmetric dpolytope must have at least 2^d vertices. 

Faces tangent to the sphere 

When  Who  What  References 
28.01.2014  Hao Chen  How to cage an egg.Every 3dimensional polytope can be realized with all its edges tangent to a sphere. This surprising result remains true if a sphere is replaced for any convex body in R^3 who is nice enough (smooth strictly convex body). If, in addition, the boundary of C satisfies stronger smoothness conditions, the space of all such polytopes Q forms a sixdimensional manifold. 

28.01.2014  Hao Chen  Hypereggs cannot be caged.This phenomenon of 3polytopes (existence of realizations with edges tangent to a sphere) is very special since for every pair (m,d)!=(1,3) there is a dpolytope that cannot be realized with all its mfaces tangent to a sphere. 

4.02.2014  Philip Brinkmann  Inscribability and Delaunay subdivisions.★Which combinatorial types of polytopes can be realized with all the vertices on the sphere? Via a stereographic projection, this question is equivalent to the study of the possible combinatorial types of Delaunay subdivisions. 

Isoperimetric quotients 

When  Who  What  References 
11.02.2014  Christoph Peters  Circumscribed polytopes.The isoperimetric quotient of a convex body C is S(C)^d/V(C)^(d1), where S and V represent the surface are and the volume, respectively. A classical result by Lindelöf states that among all polytopes with fixed facet normals, circumscribed polytopes attain the minimum isoperimetric quotient. 
