Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

Andreas Paffenholz
Technische Universität Darmstadt
Permutation Polytopes
DMV Tagung 2011 - Köln, 19. - 22. September
A permutation polytope is the convex hull of the permutation matrices of a subgroup of Sn. These polytopes
are a special class of 0/1-polytopes. A well-known example is the Birkhoff polytope of all doubly-stochastic
matrices. This polytope is defined by the full symmetric group Sn. Much less is known for general groups.
I will introduce basic properties of permutation polytopes, characterize faces, and discuss connections
between the group and the polytope. The main focus of my presentation will be on recent results for
permutation polytopes defined by cyclic groups. Their face structure depends on the cycle structure of
the generator, and I will describe exponential families of facets. The class of cyclic permutation polytopes
contains the class of marginal polytopes.
(This is joint work with Barbara Baumeister, Christian Haase, and Benjamin Nill.)
Raman Sanyal
University of California, Berkeley
Deciding Polyhedrality of Spectrahedra
Spectrahedra are to semidefinite programming what polyhedra are to linear programming. Spectrahedra
form a rich class of convex bodies with many of the favorable properties of polyhedra. It is a theoretical
interesting and practically relevant question to decide when a spectrahedron is a polyhedron. In this
talk I will discuss an algorithm for doing that which requires a good understanding of the geometry of
spectrahedra and some linear algebra. Only knowledge of the latter will be assumed.
(This is joint work with Avinash Bhardwaj and Philipp Rostalski.)
Margarita Spirova
Technische Universität Chemnitz
Translative Coverings of Convex Bodies
We discuss arrangements of proper translates of a convex body K in R n sufficient to cover this body.
We call such an arrangement a t-covering (a covering by translates) of K. First we investigate relations
between t-coverings of the whole of K and t-coverings of only its boundary. Refining the notion of tcovering
in several ways, we discuss, particularly for centrally symmetric convex bodies and n = 2, how
such coverings relate to the classical theorems of ¸Ti¸teica and Miquel as well as to notions like Voronoi
regions. We also compare t-coverings with coverings in the spirit of Hadwiger, using smaller homothetical
copies of K instead of proper translates. Finally we give upper bounds on the cardinalities of t-coverings.
(The talk is based on a joint work with Marek Lassak and Horst Martini.)
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