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ENVELOPES FOR SETS AND FUNCTIONS: REGULARIZATION AND GENERALIZED CONJUGACY

Part of: General convexity Approximations and expansions Existence theories

Published online by Cambridge University Press:  23 February 2017

A. Cabot ,
A. Jourani  and
L. Thibault
A. Cabot
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, 21000 Dijon, France email alexandre.cabot@u-bourgogne.fr
A. Jourani
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, 21000 Dijon, France email abderrahim.jourani@u-bourgogne.fr
L. Thibault
Affiliation:
Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Place EugÚne Bataillon, 34095 Montpellier, France Centro de Modelamiento MatematicoUniversidad de Chile, Chile email lionel.thibault@univ-montp2.fr
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Abstract

Let $X$ be a vector space and let $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$ be an extended real-valued function. For every function $f:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$, let us define the $\unicode[STIX]{x1D711}$-envelope of $f$ by

$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}(x)=\sup _{y\in X}\unicode[STIX]{x1D711}(x-y)\begin{array}{@{}c@{}}-\\ \cdot \end{array}f(y),\end{eqnarray}$$
where $\begin{array}{@{}c@{}}-\\ \\ \\ \cdot \end{array}$ denotes the lower subtraction in $\mathbb{R}\cup \{-\infty ,+\infty \}$. The main purpose of this paper is to study in great detail the properties of the important generalized conjugation map $f\mapsto f^{\unicode[STIX]{x1D711}}$. When the function $\unicode[STIX]{x1D711}$ is closed and convex, $\unicode[STIX]{x1D711}$-envelopes can be expressed as Legendre–Fenchel conjugates. By particularizing with $\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$, for $\unicode[STIX]{x1D706}>0$ and $p\geqslant 1$, this allows us to derive new expressions of the Klee envelopes with index $\unicode[STIX]{x1D706}$ and power $p$. Links between $\unicode[STIX]{x1D711}$-envelopes and Legendre–Fenchel conjugates are also explored when $-\unicode[STIX]{x1D711}$ is closed and convex. The case of Moreau envelopes is examined as a particular case. In addition to the $\unicode[STIX]{x1D711}$-envelopes of functions, a parallel notion of envelope is introduced for subsets of $X$. Given subsets $\unicode[STIX]{x1D6EC}$, $C\subset X$, we define the $\unicode[STIX]{x1D6EC}$-envelope of $C$ as $C^{\unicode[STIX]{x1D6EC}}=\bigcap _{x\in C}(x+\unicode[STIX]{x1D6EC})$. Connections between the transform $C\mapsto C^{\unicode[STIX]{x1D6EC}}$ and the aforestated $\unicode[STIX]{x1D711}$-conjugation are investigated.

MSC classification

Primary: 52A41: Convex functions and convex programs 49J53: Set-valued and variational analysis 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 383 - 432
Copyright
Copyright © University College London 2017 

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