Birkhoff’s version of Hilbert’s metric and its applications in analysis
Bas LemmensUniversity of Kent, Canterbury, United Kingdom
Roger D. NussbaumRutgers University, Piscataway, USA
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Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter.
Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries.