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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.

A closed half-space is a set of the form $\{x:a^{T}x\le b\}$ for a nonzero vector $a$; it is a convex set and is closed.

A compact convex set $K\subseteq {\mathbb{R}}^n$ is a compact set that is also a convex set. If $K$ is nonempty, its support function $H_K(\eta )={\max }_{x\in K}x\cdot \eta$ is finite in every direction. For such a nonempty $K$, every exterior point $x_0$ admits a strictly separating hyperplane: minimize $|x_0-y|$ over $y\in K$, and put $\omega =x_0-y$. Convexity and differentiation along the segment from $y$ to any $z\in K$ give $\omega \cdot (z-y)\le 0$, while $\omega \cdot (x_0-y)=|\omega {|}^2>0$. This is the separation used to recover spatial support from Fourier growth.

Convex analysis studies convex sets and convex functions, including epigraphs, convex conjugates and subdifferentials.

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

The core consists of efficient payoff vectors for which each coalition receives at least its own attainable value. Thus no coalition can improve every member's payoff by leaving. It is a convex set defined by linear constraints, but can be empty. Every convex cooperative game has a nonempty core because each marginal contribution vector belongs to it.

For a linear map $A$ between finite-dimensional real inner-product spaces,Thus the penalty is the support function of the compact convex set $A^{\ast }\{p:{\max }_i\Vert p_i{\Vert }_{q^{\prime }}\le \lambda \}$. The Moreau decomposition makes its proximal operator the identity minus the Euclidean projection onto a convex set.

The epigraph is $\{(u,t):E(u)\le t\}$. It is a convex set exactly when $E$ is a convex function, and is a sequentially closed set in the product topology exactly when $E$ is sequentially lower semicontinuous.

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.

A subset of a Riemannian manifold is geodesically convex when any two of its points can be joined by a minimizing geodesic segment lying in the subset. In the hyperbolic plane the joining geodesic segment is unique. Thus in the Beltrami-Klein model, geodesically convex subsets correspond to Euclidean convex sets.

A closed half-space consists of a hyperplane and one of its two sides in a real affine space. Replacing the weak inequality by a strict inequality gives an open half-space. Half-spaces are convex sets, and finite intersections of closed half-spaces are linear polyhedra.

A closed convex set in a finite-dimensional Euclidean space is the intersection of all containing closed half-spaces. For a point outside a nonempty set, its Euclidean projection onto a convex set gives a separating normal. Applied to an epigraph, this turns geometric separation into affine lower bounds and is a key step in the Fenchel-Moreau theorem.

The hit-and-run sampler moves within a full-dimensional compact convex set by choosing a uniform direction, then a uniform point on the chord through the current point in that direction. Its invariant law is the uniform distribution on the set. Starting in the interior keeps the chain in the interior almost surely. To define absolutely continuous transitions also at boundary points, redraw directions whose chords have zero length. Without this convention, a corner can have a positive holding atom.

An information projection of $Q$ onto a set $\mathcal{C}$ of probability distributions minimizes $D(P\Vert Q)$ over $P\in \mathcal{C}$. For a closed convex set and suitable $Q$, it obeys the Pythagorean inequality

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.