Polarities and Generalized Extremal Convolutions

Abstract : A reflection on the inceptive role of Moreau in various studies is presented. Gustave Choquet (1915-2006), Jean Jacques Moreau (1923-2014) and Ennio De Giorgi (1928-1996) inspired developments of variational analysis. I had the fortune to know them all and to follow their tracks. Jean Jacques Moreau presented in [18] and [19, p. 122] an abstract duality scheme of hulls with respect to arbitrary couplings, rather than the bilinear coupling used by Fenchel [12]. Moreau's scheme inspired Stanislaw Kurcyusz and myself in [8, 9] in our study of generalized Lagrangeans. Investigations of stability of duality schemes lead Gabriele Greco and me to convergence theory [4, 5], founded by Gustave Choquet in [2], and later to the theory of variational convergences of Ennio De Giorgi and his School (see [13] of De Giorgi and Franzoni), theorized by Greco ([14, 16, 15]). The inceptive role of Moreau was consequential in my subsequent mathematical activity. Moreau conjugates were a starting point to my paper with Kurcyusz [9] and his extremal convolutions to that with Guillerme, Lignola and Malivert [7] on stability of generalized extremal convolutions(1) E-phi(alpha) (f, g) := ext(X)(alpha)phi(f, g), (1) of f, g is an element of(R) over bar (XxY), of which Moreau conjugates and inf-convolutions(2) constitute 1 Here phi : (R) over bar x (R) over bar -> (R) over bar, alpha is an element of {-1, +1}, h : X x Y -> (R) over bar and ext(X)(-1) h := inf(x is an element of X) h(x, y) and ext(X)(+1) h := sup(x is an element of X) h(x, y). 2. The inf-convolution of f, h is an element of (R) over bar (X) where X is a vector space, defined by (f del h) (y) := inf(x is an element of X) (f (x) + h (y-x)) becomes (1) if g (x, y) = h (y - x), alpha = -1 and phi (r, s) := r (+)over dot s. Here (+)over dot is the (commutative) Moreau upper extension of the addition such that (+infinity) (+)over dot (-infinity) = +infinity. special cases. The latter paper has been largely unnoticed, though it remains actual, probably partly because of its quite abstract appearance. Let me recall some of its main ideas.
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Contributeur : Imb - Université de Bourgogne <>
Soumis le : mardi 6 décembre 2016 - 16:53:51
Dernière modification le : mercredi 7 décembre 2016 - 09:16:25


  • HAL Id : hal-01410535, version 1



Szymon Dolecki,. Polarities and Generalized Extremal Convolutions. Journal of Convex Analysis, Heldermann, 2016, 23 (2), pp.603-614. <https://apps.webofknowledge.com/full_record.do?product=WOS&search_mode=GeneralSearch&qid=6&SID=W15CPGkqjVcwoW4QhdT&page=1&doc=1>. <hal-01410535>



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