If X is a complete metric space, the collection of all non-empty compact subsets of X forms a complete metric space (H(X), h), where h is the Hausdorff metric. In this paper we explore some of the geometry of the space H(Rn). Specifically, we concentrate on understanding lines in H(R). In particular, we show that for any two points A, B, ∈ H(Rn), there exist infinitely many points on the line joining A and B. We characterize some points on the lines formed using closed and bounded intervals of R and show that two distinct lines in H(R) can intersect in infinitely many points.
Faculty Advisor Name
Faculty Advisor Institution
Grand Valley State University
Suggested Mathematics Subject Classification(s)
Christopher Frayer, Some Geometry of H(Rn), Furman University Electronic Journal of Undergraduate Mathematics, 8 (2016), 1-13. Available at: http://scholarexchange.furman.edu/fuejum/vol8/iss1/1