Worksheet 6-2: Convex Cones

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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.

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

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

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

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(d) The region shown below:

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

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

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

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