### Sidebar

seminars:comb:abstract.200710jun

# A Simplified Proof for Duality of the Euclidean Property in Oriented Matroids

## Abstract for the Combinatorics Seminar 2007 October 2

It is a well-known fact that the dual of a Euclidean oriented matroid program is Euclidean. The original proof, mimicking the Simplex Algorithm, uses pivot steps and tableaux to show that there is a correspondence between non-Euclidean cycles in the original program and non-Euclidean cycles in the dual. I will simplify this proof. In my proof I will work with cycles in a pseudo-sphere representation of the oriented matroid together with labels on their vertices. Such a cycle will be called a labeled cycle. The proof goes by proving that there is a bijection between non-Euclidean labeled cycles in the original program and non-Euclidean labeled cycles in the dual. This bijection is quite simple and it has a very explicit formula.