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The book presents a systematic and unified study of geometric
nonlinear functional analysis. This area has its classical roots in
the beginning of the twentieth century and is now a very active
research area, having close connections to geometric measure
theory, probability, classical analysis, combinatorics, and Banach
space theory. The main theme of the book is the study of uniformly
continuous and Lipschitz functions between Banach spaces (e.g.,
differentiability, stability, approximation, existence of
extensions, fixed points, etc.). This study leads naturally also to
the classification of Banach spaces and of their important subsets
(mainly spheres) in the uniform and Lipschitz categories. Many
recent rather deep theorems and delicate examples are included with
complete and detailed proofs. Challenging open problems are
described and explained, and promising new research directions are
indicated.
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