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For a probability measure whose identity map is strongly measurable with integrable norm, its barycenter is the Bochner integral of that map. It is characterized by $\phi (b(\mu ))=\int \phi (x)d\mu (x)$ for every continuous linear functional. For a measure on a weakly compact set in a separable Banach space, its barycenter belongs to the norm-closed convex hull of that set.

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.

The smallest closed convex set containing a set $C$, equivalently the closure of its convex hull in the specified topology. The topology must be stated: the norm topology, weak topology and weak-star topology need not give the same closure for an arbitrary set.

In an $N$-dimensional real vector space, each member of a conic hull has a representation using at most $N$ generators. To prove it, take a finite positive-coefficient representation with more than $N$ terms. Its generators are linearly dependent, say ${\sum }_j{\mu }_j{s}_j=0$, with some ${\mu }_j>0$ after reversing the relation if necessary. Subtract $t{\mu }_j$ from each coefficient, taking $t={\min }_{{\mu }_j>0}{\lambda }_j/{\mu }_j$. All coefficients stay nonnegative and at least one vanishes. Iterate. The bound differs from the $N+1$ bound for a convex hull because the coefficient sum is unrestricted.

The study of convex sets, their convex hulls, supporting closed half-spaces, and inequalities for their sizes.

The convex hull of the positive and negative coordinate unit vectors. Its boundary triangulates a sphere, and each face contains at most one member of each antipodal pair of coordinate unit vectors. This makes its barycentric subdivision useful for triangulating Real projective space.

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.

For a finite support whose convex hull has full dimension, every interior target mean is realized by a unique natural parameter. Minimize the coercive function $\log {\sum }_n{e}^{\theta \cdot n}-\theta \cdot \mu$; its gradient is the mean discrepancy and its Hessian matrix is the positive-definite covariance matrix of the support.

A probability distribution over joint action profiles implements a convex combination of their two-player payoff vectors. The attainable expected payoffs form the convex hull of the pure joint payoffs. The distribution need not factor into independent mixed strategies, and a freely enforceable lottery need not satisfy the incentive constraints of a Nash equilibrium.

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.