# Cone

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

# Convex cone

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

## Examples:

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

# Related definitions

## 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 $\theta_i \ge 0$

## Conic hull

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