Extreme direction of convex cone as extreme point of intersection with hyperplane

Dependencies:

1. Inner product space
2. Convex set
3. Cone
4. Extreme directions of a convex set
5. Extreme point of a convex set
6. Convex combination and convex hull
7. Inner product is anti-linear in second argument

$$ $$ Let $C$ be a convex cone (in an inner product space over $\mathbb{R}$). Let $c$ be a vector such that $⟨c,x⟩>0$ for all $x\in C-\{0\}$. Let $\gamma \in {\mathbb{R}}_{>0}$ and $S=\{x\in C:⟨c,x⟩=\gamma \}$.

If $d$ is an extreme direction of $C$, then $(\gamma /⟨c,d⟩)d$ is an extreme point of $S$. If $d$ is an extreme point of $S$, then $d$ is an extreme direction of $C$.

Proof

We can assume without loss of generality that $\gamma$ is 1, since we can scale $c$.

Two vectors $u$ and $v$ are called collinear iff $u/⟨c,u⟩=v/⟨c,v⟩$.

Let $d\in C-\{0\}$. Let $\hat{d}=d/⟨c,d⟩$. Then $⟨c,\hat{d}⟩=1$, so $\hat{d}\in S$. Suppose $\hat{d}$ is a non-extreme point of $S$. Then $\hat{d}=(1-\alpha )d^{(1)}+\alpha d^{(2)}$ for some $d^{(1)},d^{(2)}\in S$ where $d^{(1)}\ne d^{(2)}$ and $\alpha \in (0,1)$. Note that $d^{(1)}$ and $d^{(2)}$ are not collinear. Then $d=⟨c,d⟩(1-\alpha )d^{(1)}+⟨c,d⟩\alpha d^{(2)}$, so $d$ is not an extreme direction of $C$. Hence, if $d$ is an extreme direction of $C$, then $\hat{d}$ is an extreme point of $S$.

Let $d$ be an extreme point of $S$. Then $d\ne 0$. Suppose $d$ is not an extreme direction of $C$. Then $d=\alpha d^{(1)}+\beta d^{(2)}$, such that $\alpha >0$, $\beta >0$, and $d^{(1)},d^{(2)}\in C-\{0\}$ such that $d^{(1)}/⟨c,d^{(1)}⟩\ne d^{(2)}/⟨c,d^{(2)}⟩$. Let ${\hat{d}}^{(1)}=d^{(1)}/⟨c,d^{(1)}⟩$ and ${\hat{d}}^{(2)}=d^{(2)}/⟨c,d^{(2)}⟩$. Then ${\hat{d}}^{(1)}\ne {\hat{d}}^{(2)}$ and ${\hat{d}}^{(1)},{\hat{d}}^{(2)}\in S$. Hence, we get $d=(\alpha ⟨c,d^{(1)}⟩){\hat{d}}^{(1)}+(\beta ⟨c,d^{(2)}⟩){\hat{d}}^{(2)}$. Since, $1=⟨c,d⟩=\alpha ⟨c,d^{(1)}⟩+\beta ⟨c,d^{(2)}⟩$, we get that $d$ is a strict convex combination of ${\hat{d}}^{(1)}$ and ${\hat{d}}^{(2)}$. Hence, $d$ is not an extreme point of $S$, which is a contradiction. Hence, $d$ is an extreme direction of $C$.

Dependency for:

1. Representing point in pointed polyhedral cone

Info:

• Depth: 8
• Number of transitive dependencies: 13

Transitive dependencies:

1. /complex-numbers/conjugation-is-homomorphic
2. Group
3. Ring
4. Field
5. Vector Space
6. Inner product space
7. Inner product is anti-linear in second argument
8. Cone
9. Convex combination and convex hull
10. Convex set
11. Extreme point of a convex set
12. Direction of a convex set and Recession cone
13. Extreme directions of a convex set