1. Convex sets

• Definition 1: A set $\mathbf{C}$ is convex if the line segment between any two distinct points in $\mathbf{C}$ lies in $\mathbf{C}$.

  • For any $\mathbf{x_1}$, $\mathbf{x_2}$ $\in\mathbf{C}$, $\theta\in[0,1]$, we have $\theta\mathbf{x_1}+(1-\theta)\mathbf{x_2}\in\mathbf{C}$.

• With $\forall_{i=1}^k\mathbf{x}_i\in\mathbf{C}$ and $\begin{cases}\forall_{i=1}^k\theta_i\ge0\\\sum_{i=1}^k\theta_i=1\end{cases}$, we call a point of a form $\sum_{i=1}^k\theta_i\mathbf{x}_i$ is a convex combination.

• As with affine sets (mentioned in CVX_1), we can show that a set is convex if it contains every convex combination of its points. So we ignore the proof.

• Definition 2: A convex hull of a set $\mathbf{X}$, denoted $\text{conv}\mathbf{X}$ is a set of all convex combinations of points in $\mathbf{X}$.

  $$\text{conv}\mathbf{X}=\biggl\{\sum_{i=1}^k\theta_i\mathbf{x}_i \ \biggl|\ \forall_{i=1}^k\mathbf{x}_i\in\mathbf{X},\ \forall_{i=1}^k\theta_i\ge0,\ \sum_{i=1}^k\theta_i=1\biggl\}$$

2. Cones

• Definition 3: A set $\mathbf{C}$ is called cone or nonnegative homogeneous if for every $\mathbf{x}\in\mathbf{C}$ and $\theta\ge0$, we have $\theta\mathbf{x}\in\mathbf{C}$.

• Definition 4: A set $\mathbf{C}$ is convex cone if it is convex and a cone.

• For any $\mathbf{x_1}, \mathbf{x_2}\in\mathbf{C}$ and $\theta_1,\theta_2\ge0$, we have $\theta_1\mathbf{x_1}+\theta_2\mathbf{x_2}\in\mathbf{C}$.

• Similar to convex sets, we can define a point $\sum_{i=1}^k\theta_i\mathbf{x}_i$ is a conic combination with $\forall_{i=1}^k\mathbf{x}_i\in\mathbf{C}$ and $\forall_{i=1}^k\theta_i\ge0$. And again, a set is cone if it contains every conic combination of its.

• Definition 5: The conic hull of a set $\mathbf{X}$ is the set of all conic combinations of points in $\mathbf{X}$:

  $$\biggl\{\sum_{i=1}^k\theta_i\mathbf{x}_i \ \biggl|\ \forall_{i=1}^k\mathbf{x}_i\in\mathbf{X},\ \forall_{i=1}^k\theta_i\ge0\biggl\}$$

  which is also smallest convex cone that contains $\mathbf{X}$.

Reference:

• Chapter 2 | Convex Optimization.