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