= Extreme-point criterion for the L-infinity unit ball
{title2=$\operatorname{Ex}(B_{L^\infty})=\{h:|h|=1\ \text{almost everywhere}\}$}
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.
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