Inversion and Sphericalization of Metric Spaces

Metric space geometry is a field of mathematics concerned with how we measure distances on a set of points. There is the distance that everybody learns about in middle school math, that is, the euclidean measure of distance. There exist more exotic notions of distances, however. Some of these notions are very practical, like the taxi-cab distance. This is a measure of distance using the distance that a taxi would have to drive in order to get to a destination. Some of the notions of distances are yet more exotic and do not have as obvious of a use in the real world. One of these distances is called the Ferrand distance. Dr. Herron and I worked on obtaining estimates for the Ferrand distance between two points by using only properties of the ambient metric space itself. We were able to set upper and lower limits on the Ferrand distance using quantities such as the diameter of the boundary of our metric space. This work we performed was based on a previous study that Dr. Herron did, here in which he and his collaborators considered very similar questions about a different notion of distance, called the quasi-hyperbolic distance.

The senior thesis document can be found here.