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Zbl 0745.49014
Property of convexification.
(Italian. English summary)
[J] Rend. Semin. Mat. Univ. Padova 85, 55-78 (1991). ISSN 0041-8994

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, ${\cal R}\subset 2\sp Z, {\cal A}\subset\{ a: Z\to [-\infty,+\infty] \text{ functions}\}$, we say that $(g,h)$ verifies a functional estimation concerning ${\cal R},Z,{\cal A}$ (and we write $(g,h)\in M({\cal R},Z,{\cal A})$) if $g,h: Z\to [-\infty ,+\infty ]$ are functions such that there exists $Z\sb 0 \subset Z$ with $Z\setminus Z\sb 0 \in {\cal R}$ and for each $\epsilon >0$ there exists $a\sb \epsilon \in {\cal A}$ for which $h(z)\leq a\sb \epsilon (z) + \epsilon g(z)$ for every $z\in \{w\in Z\sb 0: h(w) > - \infty$, $g(w) < +\infty$, $a\sb \epsilon (w) < +\infty\}$. She gives sufficient conditions to obtain that, if a couple $(g,h)\in M({\cal R},Z,{\cal A})$, then also the couple $(g\sp{**},h\sp{**})$ of double Fenchel conjugates (with respect to the last variable) belongs to $M({\cal R},Z,{\cal A})$. 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 $\cal A$ and $\cal R$ if $f: T\times (X\times Y)\to [0,+\infty]$ is a normal integrand (sequentially lower semicontinuous) such that $(f,(t,x,y) \in T\times X\times Y \not\to \vert y\vert \sb Y \in [0,+\infty[) \in M({\cal R},T\times X\times Y,{\cal A})$ 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 $(\alpha ,\beta\circ (pr\sb T,\gamma))$ 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.
MSC 2000:
*49J52 Nonsmooth analysis (other weak concepts of optimality)

Keywords: functional estimation; double Fenchel conjugates

Citations: Zbl 0723.49004

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