Extreme point

An extreme point of a convex set $C$ is a point $x\in C$ for whichimplies $x=y=z$.

For compact Hausdorff $K$, the extreme points of the real unit ball in the space of continuous functions on a compact space are precisely its continuous sign-valued functions. A point with $|f|<1$ permits a nonzero continuous bump perturbation supported where there is uniform slack, while pointwise equality at an endpoint of $[-1,1]$ forces both functions in a midpoint decomposition to agree. For the Cantor set, clopen indicators give the perturbations directly.

Every real continuous function of supremum norm at most one on the Cantor set is a uniform limit of convex combinations of continuous sign functions. Approximate on a finite Cantor cylinder partition by values $a_j\in [-1,1]$. The sign vector $s$ has weight ${\prod }_j(1+s_j{a}_j)/2$, whose coordinate means are $a_j$. This proves a closed convex hull identity without weak compactness.

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.