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 \(L^p\)-approximation, as \(p \rightarrow +\infty \) 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 \(L^\infty \) context.