Convex cone

In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, ${\displaystyle C}$ is a cone if ${\displaystyle x\in C}$ implies ${\displaystyle sx\in C}$ for every positive scalar ${\displaystyle s}$. This is a broad generalization of the standard cone in Euclidean space.

A convex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets.

The definition of a convex cone makes sense in a vector space over any ordered field, although the field of real numbers is used most often.

A subset ${\displaystyle C}$ of a vector space is a cone if ${\displaystyle x\in C}$ implies ${\displaystyle sx\in C}$ for every ${\displaystyle s>0}$. Here ${\displaystyle s>0}$ refers to (strict) positivity in the scalar field.

Some other authors require ${\displaystyle [0,\infty )C\subset C}$ or even ${\displaystyle 0\in C}$. Some require a cone to be convex and/or satisfy ${\displaystyle C\cap -C\subset \{0\}}$.

The conical hull of a set ${\displaystyle C}$ is defined as the smallest convex cone that contains ${\displaystyle C\cup \{0\}}$. Therefore, it need not be the smallest cone that contains ${\displaystyle C\cup \{0\}}$.

Wedge may refer to what we call cones (when "cone" is reserved for something stronger), or just to a subset of them, depending on the author.

A subset ${\displaystyle C}$ of a vector space ${\displaystyle V}$ over an ordered field ${\displaystyle F}$ is a cone (or sometimes called a linear cone) if for each ${\displaystyle x}$ in ${\displaystyle C}$ and positive scalar ${\displaystyle \alpha }$ in ${\displaystyle F}$, the product ${\displaystyle \alpha x}$ is in ${\displaystyle C}$. Note that some authors define cone with the scalar ${\displaystyle \alpha }$ ranging over all non-negative scalars (rather than all positive scalars, which does not include 0). Some authors even require ${\displaystyle 0\in C}$, thus excluding the empty set.