Master of Science
School of Mathematics and Applied Statistics
Chad, Benjamin Michael-John, Average distances in Euclidean and other spaces, Master of Science thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2005. http://ro.uow.edu.au/theses/2713
This thesis is a study of average distances in compact metric spaces. The average distances that we study can be motivated by questions such as "How can one arrange a finite set of points on a square so as to maximise the average distance between them?" The answer to this question is simple: Place a point on each vertex of the square! Such questions become much more interesting, and their corresponding answers become much harder to find, when we generalise the space to other metric spaces, and vary what we mean by the term "average distance". For instance, is it always possible given any finite set of points on the boundary of a circle to find another point in the same region such that the "average distance" from the point to the set is 2/7r? If so, will any other "magic-number" other than 2/7r work? We define three average distances for a compact metric space (the first as a result of the Gross-Stadje Theorem, which additionally requires that the space be connected, the others as suprema of certain average distances) which form our focus of study. These averages have previously been discussed in mathematical journals, and one purpose of this thesis is to survey much of the known material regarding them. Indeed, until now there has been no single account which discusses all three averages together, and we gain much further insight by doing so.