Revisited convexity notions for $L^{\mathrm{\infty }}$ variational problems

We address a deep study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals as well as those raised by the power-law approximation of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork to ease any research in the $L^{\mathrm{\infty }}$ context.