Conic set

1 Cone

A non-empty set $S$ is called a cone, if:

$$\forall x \in S, \; \theta \ge 0 \;\; \rightarrow \;\; \theta x \in S$$

Figure 1: Illustration of a cone

2 Convex cone

The set $S$ is called a convex cone, if:

$$\forall x_1, x_2 \in S, \; \theta_1, \theta_2 \ge 0 \;\; \rightarrow \;\; \theta_1 x_1 + \theta_2 x_2 \in S$$

Figure 2: Illustration of a convex cone

Example

• $\mathbb{R}^n$
• Affine sets, containing $0$
• Ray
• $\mathbf{S}^n_+$ - the set of symmetric positive semi-definite matrices

3 Related definitions

3.1 Conic combination

Let we have $x_1, x_2, \ldots, x_k \in S$, then the point $\theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_k x_k$ is called conic combination of $x_1, x_2, \ldots, x_k$ if $\theta_i \ge 0$.

3.2 Conic hull

The set of all conic combinations of points in set $S$ is called the conic hull of $S$:

$$\mathbf{cone}(S) = \left\{ \sum\limits_{i=1}^k\theta_i x_i \mid x_i \in S, \; \theta_i \ge 0\right\}$$

Figure 3: Illustration of a convex hull