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 , is a cone if

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

What the condition preserves

A cone preserves direction from the origin, not distance. If , then all stretched and shrunk versions

also lie in . 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.