Definition of Cone and Convex Cone 📂Linear Algebra

Definition of Cone and Convex Cone

Definition ¹

Cone

A cone is defined in a vector space $V$ as a subset $C \subset V$ that satisfies the following for all scalars $a > 0$ and $x \in C$: $$ ax \in C $$

Flat Cone and Salient Cone

If cone $V$ satisfies $-\mathbf{v} \in V$ for some non-zero vector $\mathbf{v} \in V$, it is called a flat cone. If not, it is called a salient cone.

Convex Cone

A cone $C \subset V$ that satisfies the following for all scalars $a, b > 0$ and $x, y \in C$ is called a convex cone: $$ ax + by \in C $$

Pointed Cone and Blunt Cone

A convex cone that includes the zero vector $\mathbf{0}$ is called a pointed cone. If not, it is called a blunt cone.

Explanation

According to the definition, all elements of a cone can be thought of as a collection of all vectors that can be scaled up or down from the zero vector $\mathbf{0}$, and the meaningful types of these vectors can be either finite or infinite without any abstract problem. For example, in Euclidean space, any half-line is a cone at $\mathbb{R}^{1}$, and the first quadrant is a cone at $\mathbb{R}^{2}$, which in fact is also a convex cone.

A cone being flat means, in simple terms, whether it has a subvector space with a dimension of at least $1$. For instance, in $\mathbb{R}^{2}$, a line parallel to the $x$-axis includes both a vector in the $x$ direction and the $-x$ direction, making it a flat cone, which aligns with the intuition of a line stretching flatly within a plane. A cone being salient means it protrudes like a geometric cone.

That a convex cone $C$ is salient is equivalent to $C \cap C = \left\{ \mathbf{0} \right\}$.

Theorem

Partial Ordering of Convex Cones

In pointed and salient convex cones $C$, a partial order $\ge \subset C^{2}$ can be defined as follows: $$ x \ge y \iff x - y \in C \qquad \forall x, y \in C $$

Proof

To prove, we need to show that the relation $\ge$ is transitive, reflexive, and antisymmetric.

(Transitive) $$ \begin{align*} & x \ge y \land y \ge z \\ \iff & x - y \in C \land y - z \in C \\ \implies & x - z = (x - y) + (y - z) \in C \\ \iff & x \ge z \end{align*} $$

(Reflexive) A cone $C$ being pointed means that the zero vector is included in $C$. $$ x \ge x \iff x - x = \mathbf{0} \in C $$

(Antisymmetric) A cone $C$ being salient means that $C$ does not include the additive inverse $-z$ of any vector $z$. $$ \begin{align*} & x \ge y \land y \ge x \\ \iff & x - y \in C \land y - x \in C \end{align*} $$ For a non-zero vector $(x-y)$, $(y-x)$ cannot be its inverse, therefore, $x = y$ must hold.

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