ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES

By Heinz H. Bauschke, Jonathan Michael Borwein, Patrick L. Combettes. 2001

319 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/73fa98ff015d8df0abcb53491564d3d69d8bc522

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Citations in the corpus

A Characterization of Bregman Firmly Nonexpansive Operators Using a New Monotonicity Concept (2011, background): “See, for instance, [2, 3, 4, 5, 6, 13, 12, 7, 14, 15, 17, 20, 21, 22, 23, 24] and the references therein.”
Right Bregman Nonexpansive Operators in Banach Spaces (2012, background): “The class of Legendre functions in infinite dimensional Banach spaces was first introduced and studied by Bauschke, Borwein and Combettes in [3].”
Uniformly convex functions on Banach spaces (2008, background): “Additionally, related properties of convex functions and their applications have been studied in papers such as [2, 3, 4, 5].”
Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces (2006, unknown): “The implications ( v ) ⇒ ( iv ) ⇒ ( iii ) ⇒ ( ii ) follow from Theorem 2.10, [28, Proposition 1.2.5] and Proposition 2.13, while the implication ( ii ) ⇒ ( i ) follows from [13, Theorem 5.11].”
Firmly Nonexpansive Mappings and Maximally Monotone Operators: Correspondence and Duality (2011, background): “0 When X is inﬁnite-dimensional, the results become more technical and additional assumptions are required due to subtleties of Legendre functions that do not occur in ﬁnite-dimensional settings; see [3] and [6]. Remark 3.11 (strong monotonicity and cocoercivity) Concerning Theorem 2.1(xi)&(xii)”
Ambiguity Aversion, Robustness, and the Variational Representation of Preferences (2006, unknown): “map from �(�) to ba(�) that associates to p(�) in �(�) the bounded and …nitely 19 This is proved in [ 4 ].”
Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces (2010, background): “Legendre functions mapping a general Banach space X into (−∞,+∞] are defined in [3].”
Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces (2010, background): “Several interesting examples of Legendre functions are presented in [3] and [4].”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, methodology): “For now, we are content to mention a powerful subdifferential formula whose proof as given in [35] relies on careful application of Borwein’s variational principle (4.”