Revisited convexity notions for \(L^\infty \) variational problems

We address a detailed study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals, as well as those arising by the \(L^p\) $L^p$ -approximation, as \(p \rightarrow +\infty \) $p\to +\infty$ of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork underlying further research in the \(L^\infty \) $L^{\infty }$ context.