Extreme-point criterion for the L-infinity unit ball ($\mathrm{Ex}(B_{L^{\infty }})=\{h:|h|=1\text{almost everywhere}\}$)

For real scalars the extremes are the almost-everywhere sign-valued functions. A positive-measure region uniformly inside $[-1,1]$ permits opposite nontrivial bounded perturbations, proving necessity. Equality at either endpoint forces both members of a convex decomposition to agree, proving sufficiency. For complex scalars the same criterion uses unit-modulus functions and the strict convexity of the complex unit disk.