# Any law of group metric invariant is an inf-convolution.

1 Equations d'evolution
SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
Abstract : In this article, we bring a new light on the concept of the inf-convolution operation $\oplus$ and provides additional informations to the work started in \cite{Ba1} and \cite{Ba2}. It is shown that any internal law of group metric invariant (even quasigroup) can be considered as an inf-convolution. Consequently, the operation of the inf-convolution of functions on a group metric invariant is in reality an extension of the internal law of $X$ to spaces of functions on $X$. We give an example of monoid $(S(X),\oplus)$ for the inf-convolution structure, (which is dense in the set of all $1$-Lipschitz bounded from bellow functions) for which, the map $\arg\min : (S(X),\oplus) \rightarrow (X,.)$ is a (single valued) monoid morphism. It is also proved that, given a group complete metric invariant $(X,d)$, the complete metric space $(\mathcal{K}(X),d_{\infty})$ of all Katetov maps from $X$ to $\R$ equiped with the inf-convolution has a natural monoid structure which provides the following fact: the group of all isometric automorphisms $Aut_{Iso}(\mathcal{K}(X))$ of the monoid $\mathcal{K}(X)$, is isomorphic to the group of all isometric automorphisms $Aut_{Iso}(X)$ of the group $X$. On the other hand, we prove that the subset $\mathcal{K}_C(X)$ of $\mathcal{K}(X)$ of convex functions on a Banach space $X$, can be endowed with a convex cone structure in which $X$ embeds isometrically as Banach space.
Type de document :
Pré-publication, Document de travail
2015

https://hal-paris1.archives-ouvertes.fr/hal-01169677
Contributeur : Mohammed Bachir <>
Soumis le : mardi 30 juin 2015 - 03:54:27
Dernière modification le : lundi 27 novembre 2017 - 14:14:02
Document(s) archivé(s) le : mercredi 16 septembre 2015 - 06:29:26

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Inf-conv-law.pdf
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• HAL Id : hal-01169677, version 1

### Citation

Mohammed Bachir. Any law of group metric invariant is an inf-convolution. . 2015. 〈hal-01169677〉

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