Proper cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

A subset C of a vector space V is a cone (or sometimes called a linear cone) if for each x in C and positive scalars α, the product αx is in C.

A cone C is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers. Also note that the scalars in the definition are positive meaning that origin does not have to belong to C. Some authors use a definition that ensures the origin belongs to C. Because of the scaling parameters α and β, cones are infinite in extent and not bounded.

If C is a convex cone, then for any positive scalar α and any x in C the vector αx = (α/2)x + (α/2)x is in C. It follows that a convex cone C is a special case of a linear cone.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set C is a convex cone if and only if "αC = C and C + C = C, for any positive scalar α.

An affine convex cone is the set resulting from applying an affine transformation to a convex cone. A common example is translating a convex cone by a point p: p+C. Technically, such transformations can produce non-cones. For example, unless p=0, p+C is not a linear cone. However, it is still called an affine convex cone.

A (linear) hyperplane is a set in the form ${\displaystyle \{x\in V\mid f(x)=c\}}$ where f is a linear functional on the vector space V. A closed half-space is a set in the form ${\displaystyle \{x\in V\mid f(x)\leq c\}{\text{ or }}\{x\in V\mid f(x)\geq c\}}$, and likewise an open half-space uses strict inequality.

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