Normality and Quasiconvex Integrands

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.