
Journal of Convex Analysis 17 (2010), No. 1, 059068 Copyright Heldermann Verlag 2010 Normality and Quasiconvex Integrands Abdessamad Amir University of Mostaganem, Faculty of Sciences, Dept. of Mathematics, 27000 Mostaganem, Algeria amir@univmosta.dz Hocine MokhtarKharroubi University of Oran, Faculty of Sciences, Dept. of Mathematics, Oran, Algeria hmkharroubi@yahoo.fr [Abstractpdf] 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) 525539; and in: Nonlinear Operators and the Calculus of Variations; Bruxelles 1975, Lecture Notes in Mathematics 543, 157207, 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. Keywords: Normal integrand, quasiconvex functions, conjugation. MSC: 26B25,49N15, 49J53 [ Fulltextpdf (140 KB)] for subscribers only. 