A convex cone is a subset $C$ of a real vector space $V$ that is closed under two operations:
- Positive scalar multiplication – for every $x\in C$ and every scalar $s>0$ (with respect to the ordered field of scalars, typically the real numbers), the product $sx$ also belongs to $C$.
- Addition – for any $x,y\in C$, the sum $x+y$ is again in $C$.
Equivalently, a convex cone is a set that is closed under all linear combinations with non‑negative coefficients: $$ \alpha x+\beta y\in C \quad\text{for all }x,y\in C\text{ and }\alpha,\beta>0 . $$
Because it is closed under addition, a convex cone is automatically a convex set. The definition makes sense in any vector space over an ordered field; the most common case is a finite‑dimensional space $\mathbb{R}^n$.
Basic properties
- Homogeneity – For any positive scalar $\alpha$, $\alpha C = C$.
- Unboundedness – Except for the trivial cone ${0}$, convex cones extend infinitely in at least one direction.
- Faces and extremal rays – A face of a convex cone is a convex sub‑cone that contains every line segment of the cone whose interior point lies in the face. One‑dimensional faces are called extremal rays.
- Dual cone – The dual of $C$ is the set $C^{}={y\in V^{}\mid \langle y,x\rangle\ge 0\ \forall x\in C}$; it is also a convex cone.
Common subclasses
| Subclass | Characterisation |
|---|---|
| Polyhedral cone | Intersection of finitely many closed half‑spaces; equivalently the conical hull of finitely many generators. |
| Finitely generated cone | Set of all non‑negative linear combinations of a finite set of vectors (the generators). |
| Closed cone | Contains all its limit points (topologically closed). |
| Pointed (salient) cone | Contains no line through the origin; i.e., $C\cap(-C)={0}$. |
| Affine convex cone | Image of a convex cone under an affine transformation (e.g., a translation $p+C$). |
Related concepts
- Cone (linear cone) – A set closed only under positive scalar multiplication; a convex cone adds closure under addition.
- Conical hull – The smallest convex cone containing a given set $S$; formally $\operatorname{cone}(S)={\sum_{i}\lambda_i x_i\mid x_i\in S,\ \lambda_i\ge0}$.
- Half‑space – Sets of the form ${x\mid f(x)\le c}$ (closed) or ${x\mid f(x)<c}$ (open) for a linear functional $f$; each is an affine convex cone.
Applications
Convex cones appear throughout mathematics and its applications, notably in:
- Optimization – Conic programming (e.g., linear, semidefinite, and second‑order cone programming) formulates constraints as membership in convex cones.
- Functional analysis – Dual cones describe positivity conditions for linear functionals.
- Geometry – Polyhedral cones model tangent cones of convex polyhedra and feasible direction sets.
- Economics and game theory – Production possibility sets and payoff cones are convex cones.
References
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Wikipedia, “Convex cone”, accessed via Jina AI mirror, 2023. Source
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Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.
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Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.