Geometry and Combinatorial Optimization
Org:
Guyslain Naves (AixMarseille University)
[
PDF]
 MARCEL CELAYA, Georgia Tech
The linear span of lattice points in the halfopen unit cube [PDF]

Let $\Lambda\subset\mathbf{R}^n$ be a lattice which contains the integer lattice $\mathbf{Z}^n$. We characterize the space of linear functions $\mathbf{R}^n\rightarrow\mathbf{R}$ which vanish on the lattice points of $\Lambda$ lying in the halfopen unit cube $[0,1)^n$. We also find an explicit formula for the dimension of the linear span of $\Lambda\cap[0,1)^n$.
 ROBERT DAVIS, Michigan State University
Detecting the Integer Decomposition Property in Reflexive Simplices [PDF]

The Ehrhart series of a lattice polytope $P$ is a rational function encoding the number of lattice points in nonnegative integer dilates of $P$. The numerator of this series is the (Ehrhart) $h^*$polynomial of $P$, and its list of coefficients, called the $h^*$vector, are nonnegative integers. A wideopen conjecture claims that all Gorenstein polytopes with the integer decomposition property (IDP) have unimodal $h^*$vectors. Without requiring IDP, the conjecture is false even in the case of reflexive simplices. In this talk, we recast the conjecture for reflexive simplices in the language of number theory according to a classification via arithmetic sequences. We then prove the conjecture for certain families of reflexive simplices, and see that there exist simplices within these families that meet only a necessary (but not sufficient) condition for possessing IDP.
 GUYSLAIN NAVES, Marseille University
Packing and covering with balls on Busemann surfaces [PDF]

We prove that for any compact subset $S$ of a Busemann surface
$({\mathcal S},d)$ (in particular, for any simple polygon with
geodesic metric) and any positive number $\delta$, the minimum number
of closed balls of radius $\delta$ with centers at $\mathcal S$ and
covering the set $S$ is at most 19 times the maximum number of
disjoint closed balls of radius $\delta$ centered at points of $S$:
$\nu(S)\le \rho(S)\le 19\nu(S)$, where $\rho(S)$ and $\nu(S)$ are the
covering and the packing numbers of $S$ by ${\delta}$balls.
 ANDRAS SEBO, Grenoble
Tours, Colouring or Somewhere In Between [PDF]

TBA