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For a doubly stochastic matrix with a fractional entry, form the row-column bipartite graph of entries strictly between zero and one. Every incident vertex has degree at least two, so there is an even cycle. A matrix supported on that cycle with alternating entries $+1,-1$ has zero row and column sums. A sufficiently small positive multiple can be both added and subtracted while preserving entry bounds. The original matrix is then the midpoint of two distinct doubly stochastic matrices, so it is not an extreme point.

Every doubly stochastic matrix is a convex combination of permutation matrices. Equivalently, the extreme points of the doubly stochastic convex polytope are exactly the permutation matrices. The extreme-point assertion follows from an alternating-cycle perturbation of a doubly stochastic matrix; the convex-hull assertion follows because a bounded finite-dimensional polytope is the convex hull of its vertices. Permutation matrices are extreme since their zero entries force the same zeros in every convex decomposition.

For an integer $0\le k\le n$, the capped simplexis the convex hull of the zero-one vectors having exactly $k$ entries equal to one. It is a convex polytope whose extreme points are precisely these vectors.

If a feasible vector has a fractional coordinate, the integer coordinate sum forces at least two fractional coordinates. Perturbing one upward and the other downward by a sufficiently small amount gives two distinct feasible vectors whose midpoint is the original vector. It is therefore not an extreme point. Conversely, a zero-one vector cannot be a nontrivial convex combination of points of $[0,1{]}^n$, since each coordinate is already at a bound.

Every point of a compact metrizable convex subset of a Hausdorff locally convex space is the barycenter of a Borel probability measure concentrated on its extreme points. The theorem is an existence assertion; uniqueness requires additional hypotheses such as a simplex structure.

For complex $C(K)$ on a compact Hausdorff space, the extreme points of its dual unit ball are precisely unimodular multiples of Dirac measures. A measure whose variation measure has mass on two disjoint sets splits as a nontrivial convex combination of normalized restrictions and is not extreme. This description is the key to the Banach–Stone theorem.

A convex set $F\subseteq C$ is a face of $C$ if $ty+(1-t)z\in F$, with $y,z\in C$ and $0<t<1$, implies $y,z\in F$. A singleton face is an extreme point. A face of a face is a face of the original convex set.

For an integer $0\le k\le n$, the fantope is the convex hull of rank-$k$ orthogonal projection matrices in ${\mathbb{R}}^n$. Equivalently, it consists of real symmetric matrices with eigenvalues in $[0,1]$ and matrix trace $k$. To prove the equivalence, apply the spectral theorem for real symmetric matrices and express the eigenvalue vector as a convex combination of the zero-one extreme points of the capped simplex. The sum of the largest eigenvalues is its support function.

If a compact convex set is the closed convex hull of a subset $S$, every extreme point lies in the closure of $S$. A neighborhood avoiding $S$ around a putative missing extreme point gives finitely many closed convex caps covering $S$ and excluding that point. The convex hull of a finite union of compact convex sets is compact: group the terms by their set and use the simplex parametrization. The point must lie in that hull, contrary to extremality. The closures are in the given Hausdorff locally convex topology.

A nonempty compact face of a convex set in a Hausdorff locally convex space contains an extreme point. Chains of compact faces have nonempty compact intersection. A minimal face supplied by the Zorn lemma must be a singleton, since a separating linear functional would otherwise expose a proper maximizer face. This is the existence step in the Krein-Milman theorem.