#
The Upcoming Monday Lecture and Colloquium

**Monday, July 14, 2008 **

Technische Universität Berlin

Fak. II, Institut für Mathematik

Str. des 17. Juni 136

10623 Berlin

room MA 041

** Lecture - 14:15**

###
Imre Bárány - Budapest and London

### Extremal problems for convex lattice polytopes

*Abstract:*

In this survey talk I will present several extremal
problems, and some solutions, concerning convex lattice polytopes.
A typical example is to determine the smallest area that a convex
lattice polygon can have if it has exactly n vertices.

**Colloquium - 16:00**

###
Eyal Ackerman - Technicon-Israel Institute of Technology, Haifa

### Improved upper bounds on the reflexivity of point sets

*Abstract:*

Given a set of $n$ points in the plane, its reflexivity is the minimum
number of reflex vertices in a simple polygonalization of the set of
points. It is conjectured that the reflexivity of any set of $n$
points is at most $n/4$. We show a $3n/7+O(1)$ upper bound, that
improves the previously known $n/2$ upper bound. Using computer-aided
abstract order type extension the upper bound can be further improved
to $5n/12+O(1)$. We also present an algorithm to compute
polygonalizations with at most this number of reflex vertices in $O(n
\log n)$ time.
joint work with Oswin Aichholzer and Balázs Keszegh.