Cone

analysis linear-algebra

Definition

Cone

A cone is a subset of a real vector space that is closed under multiplication by non-negative scalars. Formally, for $C\subseteq V$, $C$ is a cone if

$$\begin{gathered} x\in C, \\ \alpha \in {\mathbb{R}}_{\ge 0}⟹\alpha x\in C. \end{gathered}$$

Thus, once a cone contains a vector $x$, it contains the whole ray from the origin through $x$. The definition fixes the origin as the scaling centre: shrinking by $0$ sends every contained vector to $0$, so every non-empty cone contains the zero vector.

What the condition preserves

A cone preserves direction from the origin, not distance. If $x\in C$, then all stretched and shrunk versions

$$0,\frac{1}{2}x,x,2x,3x,\dots$$

also lie in $C$. What matters is membership of a ray. The absolute length of a vector is irrelevant once its direction has entered the cone.

This is why cones naturally describe directional constraints: allowed velocities, feasible improvement directions, positive combinations of generators, and local tangent directions.