# Worksheet 6-2: Convex Cones

Download: [CMSE382-WS6_2.pdf](CMSE382-WS6_2.pdf)

```{warning}
This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.
```

---

## Worksheet 6-2: Q1

For each of the following sets, determine whether it is a **cone**, a **convex cone**, both, or neither.

Recall: A set $C$ is a **cone** if $\mathbf{x} \in C \Rightarrow t\mathbf{x} \in C$ for all $t \ge 0$. It is a **convex cone** if it is both a cone and convex.

---

**(a)** $S = \{(x, |x|) \mid x \in \mathbb{R}\}$

```{image} ../../../figures/cones_question-a.png
:width: 250px
:align: center
```

---

**(b)** $S = \{(x,y) \in \mathbb{R}^2_+ \mid y \le m_1 x \text{ and } y \ge m_2 x\}$ for some $m_1 > m_2 \ge 0$

```{image} ../../../figures/cones_question-b.png
:width: 250px
:align: center
```

---

**(c)** $S = \{(x,y) \mid x \le 0, y \ge 0\} \cup \{(x,y) \mid x \ge 0, y \le 0\}$

```{image} ../../../figures/cones_question-c.png
:width: 250px
:align: center
```

---

**(d)** The region shown below:

```{image} ../../../figures/cones_question-d.png
:width: 250px
:align: center
```

---

**(e)** The Lorentz (second-order) cone:

$$S = \{(x_1, x_2, y) \in \mathbb{R}^3 \mid \|(x_1, x_2)\|_2 \le y\}$$

```{image} ../../../figures/Lorentz_cone.png
:width: 250px
:align: center
```

---

**(f)** $S = \{(x,y) \mid y \ge |x|\}$

---

**(g)** $S = \{\mathbf{0}\} \subset \mathbb{R}^n$ (the set containing only the zero vector)

```{image} ../../../figures/cones_question-f.png
:width: 150px
:align: center
```