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Property of convexification. (Italian. English summary) Zbl 0745.49014

The author studies some properties of functional estimations [introduced by himself in Rend. Acad. Naz. Sci. Detta XL, V. Ser. Mem. Math. 14, No. 1, 17-56 (1990; Zbl 0723.49004)]: if \(Z\) is a set, \({\mathcal R}\subset 2^ Z, {\mathcal A}\subset\{ a: Z\to [-\infty,+\infty] \text{ functions}\}\), we say that \((g,h)\) verifies a functional estimation concerning \({\mathcal R},Z,{\mathcal A}\) (and we write \((g,h)\in M({\mathcal R},Z,{\mathcal A})\)) if \(g,h: Z\to [-\infty ,+\infty ]\) are functions such that there exists \(Z_ 0 \subset Z\) with \(Z\setminus Z_ 0 \in {\mathcal R}\) and for each \(\epsilon >0\) there exists \(a_ \epsilon \in {\mathcal A}\) for which \(h(z)\leq a_ \epsilon (z) + \epsilon g(z)\) for every \(z\in \{w\in Z_ 0: h(w) > - \infty\), \(g(w) < +\infty\), \(a_ \epsilon (w) < +\infty\}\). She gives sufficient conditions to obtain that, if a couple \((g,h)\in M({\mathcal R},Z,{\mathcal A})\), then also the couple \((g^{**},h^{**})\) of double Fenchel conjugates (with respect to the last variable) belongs to \(M({\mathcal R},Z,{\mathcal A})\). Moreover, using the previous theorem and by means of results about a property (of type \((Q)\) of Cesari) on the convex hull of the epigraphs of sequentially lower semicontinuous functions defined on \(T\times E\times V\) which, with the “norm” on \(V\), verifies a suitable functional estimation (where \(T\) and \(E\) are pseudo-metric spaces, \(V\) is a reflexive Banach space), it is established that under suitable hypotheses on the spaces and on \(\mathcal A\) and \(\mathcal R\) if \(f: T\times (X\times Y)\to [0,+\infty]\) is a normal integrand (sequentially lower semicontinuous) such that \((f,(t,x,y) \in T\times X\times Y \not\to | y| _ Y \in [0,+\infty[) \in M({\mathcal R},T\times X\times Y,{\mathcal A})\) then the double Fenchel conjugate with respect to the last variable of \(f\) is again a normal integrand that verifies the same functional estimation. At last a hypothesis is given to prove that, if \((\alpha ,\beta\circ (pr_ T,\gamma))\) is a couple of functions defined on \(T\times X\) that satisfies a suitable functional estimation, then \((\psi ,\beta)\) verifies an analogue functional estimation, where we define as value of \(\psi: T\times W\to [-\infty ,+\infty]\) in every point the least lower bound of \(\alpha\) on suitable sets or \(\psi\) is the double Fenchel conjugate with respect to the last variable of such a least lower bound; such a double Fenchel conjugate is proved to be a normal integrand too.
Reviewer: Ada Bottaro Aruffo

MSC:

49J52 Nonsmooth analysis

Citations:

Zbl 0723.49004
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References:

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