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Essentially Smooth Lipschitz Functions

โœ Scribed by Jonathan M. Borwein; Warren B. Moors

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
535 KB
Volume
149
Category
Article
ISSN
0022-1236

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โœฆ Synopsis

In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipschitz functions that are integrable, D-representable and possess desirable differentiability properties. We also present some striking applications of our results to distance functions.

1997 Academic Press (P 3 ) f possesses differentiability properties similar to those enjoyed by continuous convex functions.

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In this note we analyze the question: When does a function f : | d ร„ | essentially depend on at most one coordinate? (Here |=N \_ [0].) For example, the function depends on both variables. However, we can cover its domain by two rectangles, 2|\_| and (2|+1)\_|, such that f depends on at most one va