Essentially smooth Lipschitz functions
type: Item
instance of: scholarly article
title: Essentially smooth Lipschitz functions
zbMATH Open document ID: 0896.49007
author: Jonathan M. Borwein
author: Warren B. Moors
published in: Journal of Functional Analysis
publication date: 1998-09-27T00:00:00Z
review text: The authors observe fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on Banach spaces. So the following 3 significant properties are discussed: (P1) \(D\)-representability, which means Gâteaux-differentiability on a dense subset \(D\) of the domain and the description of the Clarke subdifferential mapping by the gradients on \(D\); (P2) Integrability, which means the unique determination of the function (up to an additive constant) by means of its Clarke subdifferential mapping; (P3) Differentiability properties similar to those enjoyed by continuous convex functions. The aim of the paper is the examination of the relationships between these properties and the construction of a family of functions which satisfy all the three properties. After a short survey of uscos (compact-valued upper semicontinuous mappings), cuscos (convex-compact-valued upper semicontinuous mappings), and especially of minimal uscos and minimal cuscos, it is pointed out that the \(D\)-representability of a function is closely connected with the minimality of the Clarke subdifferential mapping. Moreover, it is shown that for locally Lipschitz functions which are densely Gâteaux differentiable (this is fulfilled, e.g., in separable Banach spaces) both notions are equivalent. In the main section of the paper the authors observe the class of the so-called essentially smooth functions, i.e., the class of functions which are strictly differentiable everywhere except possibly on a Haar-null set. It is shown that (in separable Banach spaces) these functions satisfy all the above mentioned properties. Moreover, this class of functions possesses very strong closure properties (e.g., regarding addition, subtraction, multiplication, division, and lattice operations) and contains many other important subclasses of Lipschitz functions (e.g., pseudo-regular and semi-smooth functions). The results are applied especially to perturbation functions and to dis…
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 1078818
zbMATH Keywords: minimal Clarke subdifferentials
zbMATH Keywords: compact-valued upper semicontinuous mappings
zbMATH Keywords: convex-compact-valued upper semicontinuous mappings
zbMATH Keywords: \(D\)-representability
zbMATH Keywords: locally Lipschitz functions
zbMATH Keywords: Gâteaux-differentiability
zbMATH Keywords: uscos
zbMATH Keywords: cuscos
zbMATH Keywords: essentially smooth functions
zbMATH Keywords: semi-smooth functions
reviewed by: Jörg Thierfelder
Wikidata QID: Q56935851
MaRDI profile type: Publication
full work available at URL: jfan.1997.3101
OpenAlex ID: W1984940349
cites work: Q3324260
cites work: Q3976070
cites work: Existence Of Nearest Points In Banach Spaces
cites work: The differentiability of real functions on normed linear space using generalized subgradients
cites work: Minimal Convex Uscos and Monotone Operators on Small Sets
cites work: Q4698683
cites work: A Chain Rule for Essentially Smooth Lipschitz Functions
cites work: Null Sets and Essentially Smooth Lipschitz Functions
cites work: Lipschitz functions with prescribed derivatives and subderivatives
cites work: Distinct differentiable functions may share the same Clarke subdifferential at all points
cites work: On sets of Haar measure zero in abelian Polish groups
cites work: Q5181909
cites work: Theorems of Namioka and R. E. Johnson Type for Upper Semicontinuous and Compact Valued Set-Valued Mappings
cites work: Optimization and nonsmooth analysis
cites work: Tangencially continuous directional derivatives in nonsmooth analysis
cites work: Subdifferential analysis of bivariate separately regular functions
cites work: Q4339230
cites work: Q3808343
cites work: Submonotone mappings in Banach spaces and applications
cites work: Q3946618
cites work: Q3946604
cites work: Generic differentiability of convex functions on the dual of a Banach space
cites work: A continuity property related to Kuratowski's index of non-compactness, its relevance to the drop property, and its implications for differentiability theory
cites work: Q4380505
cites work: D-Representation of Subdifferentials of Directionally Lipschitz Functions
cites work: Q3784624
cites work: A generic factorization theorem
cites work: Q3907986
cites work: Gateaux Differentiability of Convex Functions on Banach Spaces
cites work: A characterization of reflexive Banach spaces
cites work: Semismooth and Semiconvex Functions in Constrained Optimization
cites work: A characterisation of minimal subdifferential mappings of locally Lipschitz functions
cites work: Convex functions, monotone operators and differentiability.
cites work: Gaussian null sets and differentiability of Lipschitz map on Banach spaces
cites work: Integration of subdifferentials of nonconvex functions
cites work: Differentiability of Lipschitz functions on Banach spaces
cites work: Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings
cites work: The maximal normal operator space and integration of subdifferentials of nonconvex functions
cites work: Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces
cites work: Q3901719
cites work: On generalized differentials and subdifferentials of Lipschitz vector-valued functions
cites work: Q5601692
cites work: Q5567356
DOI: 10.1006/JFAN.1997.3101
description: scientific article; zbMATH DE number 1078818