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Metric Geometry

In: New Analytic and Geometric Methods in Inverse Problems

Author

Listed:
  • Yuri Burago

    (Steklov Institute for Mathematics at St.)

  • David Shoenthal

    (Pennsylvania State University, Department of Mathematics)

Abstract

Much of one’s mathematical experience with regard to metric spaces begins at the level of the metric. Instead of starting with a metric, in many cases we must begin with the length of paths as the primary notion. From this, we will derive a distance function. More precisely, we can introduce a new distance which is measured along the shortest path between two points in a space (as opposed to simply measuring the Euclidean distance between the two points). One says that a distance function on a metric space is an intrinsic metric if the distance between two points can be realized by paths connecting the points (mathematically, it must be equal to the infimum of lengths of paths between the points—a shortest path may not exist). If the length of paths is to be our primary notion, we must ask for a rigorous definition, from where it may arise, and what the properties are of such structures (which we will call length structures).

Suggested Citation

  • Yuri Burago & David Shoenthal, 2004. "Metric Geometry," Springer Books, in: Kenrick Bingham & Yaroslav V. Kurylev & Erkki Somersalo (ed.), New Analytic and Geometric Methods in Inverse Problems, pages 3-50, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-08966-8_1
    DOI: 10.1007/978-3-662-08966-8_1
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