S. Tsujie (Department of Information Design, Hiroshima Kokusai Gakuin University)

Collections of Unit Ball Graphs and Embeddable Finite Metric Spaces

In this talk we consider two kinds of collections of finite objects regarding metric spaces. 
One is a collection of unit ball graphs on a metric space. 
A unit ball graph is an intersection graph of balls of the same size. 
If the metric space is geodesic, then combinatorial properties of the unit ball graphs should describe geometric properties of the metric space. 
We give conditions for which the space is an $\mathbb{R}$-tree and a 1-dimensional manifold. 
The other is a collection of finite metric spaces isometrically embeddable into a metric space. 
We see that the collection characterizes the metric space when it has good properties. 
This talk is based on a joint work with Masamichi Kuroda.