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
→
+
∞
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
∞
context.