A convex cone is a subset C of a real vector space V such that for any x,y in C and any nonnegative scalars α,β ≥ 0, the linear combination αx + βy lies in C; equivalently C is closed under nonnegative linear combinations and contains the zero vector.

Principle

The organizing rule is closure under scaling by nonnegative scalars and addition, which distinguishes cones from general convex sets by allowing arbitrary nonnegative homogeneity.

Demonstration

Example: in R^n the positive orthant {x ∈ R^n : x_i ≥ 0 for all i} is a convex cone; more generally the cone generated by vectors v1,...,vk is {∑_{i} λ_i v_i : λ_i ≥ 0}.

Misapplication

Treating a set closed under convex combinations but not under arbitrary nonnegative scaling (e.g., a convex compact set that does not contain rays) as a convex cone is incorrect.

Consequence

When correctly identified, a convex cone permits use of cone-specific duality, separation theorems, and homogeneous scaling arguments; extreme rays and face lattices become meaningful invariants.

Reversal

The reversal is a linear subspace: a set closed under all real scalars (including negatives) where homogeneity allows cancellation and two-sided scaling, which removes the cone-specific ordering.

Boundary

Scope: real vector spaces (often finite-dimensional); excludes sets only closed under convex combinations but not nonnegative scaling, and does not require pointedness or closedness unless specified.

Semantic Tension

Tension arises between 'convex cone' and 'convex set': both share convexity, but cones enforce nonnegative homogeneity; confusing them obscures unbounded ray structure and dual cone properties.

Synthesis

A convex cone is a convex set with one-sided linear homogeneity: it is closed under addition and scaling by nonnegative numbers, giving a natural partial ordering, generating rays, and enabling duality constructions specific to nonnegative linear combinations.