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Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. (English) Zbl 1032.49025
As the authors mention, “the objective in this paper is to extend the classical notions of essential smoothness, essential strict convexity, and Legendreness from Euclidean to Banach spaces, to furnish an elegant and effective concomitant theory, and to demonstrate the applicability of these new notions”. The authors say that the proper lower semicontinuous convex function $f$ defined on the Banach space $X$ is essentially smooth if the Fenchel subdifferential $\partial f$ is both locally bounded and single-valued on its domain; $f$ is essentially strictly convex if ${\left(\partial f\right)}^{-1}$ is locally bounded on its domain and $f$ is strictly convex on every convex subset of $\text{dom}\phantom{\rule{4.pt}{0ex}}\partial f$; $f$ is Legendre if it is both essentially smooth and essentially strictly convex. They show that these notions reduce to the respective notions in finite dimensions. In reflexive spaces $f$ is Legendre iff ${f}^{*}$ is. They give characterizations of essential smoothness and prove that Legendre functions are zone consistent in reflexive spaces.

##### MSC:
 49J53 Set-valued and variational analysis 46G05 Derivatives, etc. (functional analysis) 52A41 Convex functions and convex programs (convex geometry) 49J50 Fréchet and Gateaux differentiability 90C25 Convex programming 46N10 Applications of functional analysis in optimization and programming