Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Barbara Langfeld
Christian-Albrechts-Universität zu Kiel
Are Planar Lattice-Convex Sets Determined by Their Covariogram?
A finite subset K of Z d is said to be lattice-convex if K is the intersection of Z d with a convex set. The
covariogram gK of K ⊆ Z d is the function associating to each u ∈ Z d the cardinality of K ∩ (K + u). Daurat,
Gérard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction
of lattice-convex sets K from gK. We provide a partial positive answer to this problem by showing
that for d = 2 and under mild extra assumptions, gK determines K up to translations and reflections. As
a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar
lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of
counterexamples.
(This is joint work with Gennadiy Averkov.)
Eva Linke
Otto-von-Guericke-Universität Magdeburg
Rational Ehrhart Quasi-Polynomials
Ehrhart’s famous theorem states that the number of integral points in a rational polytope is a quasipolynomial
in the integral dilation factor. We study the case of rational dilation factors. It turns out that
the number of integral points can still be written as a rational quasi-polynomial, that is, a polynomial
function, whose coefficients are themselves periodic functions. Furthermore, the coefficients of this
rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation. In a
special setting, the minimal periods of these coefficients are monotonically decreasing. This is not true in
the integral case, and thus we suspect that the rational quasi-polynomial preserves more of the geometric
structure of a polytope than the integral one.
Tim Netzer
Universität Leipzig
Spectrahedra
Spectrahedra are generalizations of polyhedra. They occur naturally as feasible sets, when one passes
from linear programming to semidefinite programming. Spectrahedra form a most interesting class of
sets. Most work on them is quite recent, and there are still many unsolved problems. I will give a short
introduction to the topic, and explain some of the underlying algebra.
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