Geodesic
Normality and Quasiconvex Integrands
Journal of convex analysis, Tome 17 (2010) no. 1, pp. 59-68.

Voir la notice de l'article provenant de la source Heldermann Verlag

Let $(T, \mathcal{A})$ be an arbitrary measurable space and $f$ an integrand defined on $T\times \mathbb{R}^n$ such that $f(t, \cdot)$ is quasiconvex and lower semicontinuous. Here, convexity is present by the level set mapping. We show that the normality property of the integrand in the sense of R. T. Rockafellar [Pacific Journal of Mathematics 24 (1968) 525--539; and in: Nonlinear Operators and the Calculus of Variations; Bruxelles 1975, Lecture Notes in Mathematics 543, 157--207, Springer, Berlin] can be characterized by the normality of the level set mapping, and that normality is preserved for quasiconvex conjugates. Finally we obtain for the integral $I_f (x(\cdot)) = \int_T f(t, x(t)) d\mu (t)$ the equality (in appropriate topology) between the lower semicontinuous regularization and the second quasiconvex conjugate.
Classification : 26B25,49N15, 49J53
Mots-clés : Normal integrand, quasiconvex functions, conjugation 
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     title = {Normality and {Quasiconvex} {Integrands}},
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A. Amir; H. Mokhtar-Kharroubi. Normality and Quasiconvex Integrands. Journal of convex analysis, Tome 17 (2010) no. 1, pp. 59-68. http://geodesic.mathdoc.fr/item/JCA_2010_17_1_JCA_2010_17_1_a4/