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ENVELOPES FOR SETS AND FUNCTIONS: REGULARIZATION AND GENERALIZED CONJUGACY

  • A. Cabot (a1), A. Jourani (a2) and L. Thibault (a3) (a4)

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.

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