Recession cone characterization.

Question

Let $\mathbb{C} \subset \mathbb{R^n}$ be a convex, closed and not-bounded set. Let $d$ be a vector which $||d||=1$, then show that:

$d \in recc(\mathbb{C})$ $\iff$ $\exists$ $\{x_k\}_{k\in\mathbb{N}}\subset\mathbb{C}$ which $\lim\limits_{k \longrightarrow \infty}||x_k||=\infty$ and $\lim\limits_{k \longrightarrow \infty}(x_k/||x_k||)=d$

I am using the first definition of recession cone found here Recession cone.

I've done this implication $"\Leftarrow"$, but I can't get the another one.

What does $(x_k/=||x_k||)$ mean? Typo? – Maksim Apr 15 '19 at 19:32 — Maksim, Apr 15 '19 at 19:32
@Maksim It was a mistakw, edited. – Lecter Apr 15 '19 at 21:16 — Lecter, Apr 15 '19 at 21:16

score 0 · Answer 1 · answered Apr 15 '19 at 21:39

0

If '$\Rightarrow$' would be true, since $$\lim_{k \to \infty}(x_k/||x_k||)=d$$ any $d$ would be the limit of elements with norm 1 ($||\frac{ x_k}{||x_k||}|| =1$), hence this will hold for the limit as well : $||d||=1 $.

But this is a contradiction, since in general the cone will not consist just of elements with norm $1$.

So '$\Rightarrow$' is not true.

Maybe there is something else missing ?

answered Apr 15 '19 at 21:39

Maksim

1,706

You are right. I added a new hypotheis – Lecter Apr 16 '19 at 09:58

Recession cone characterization.

1 Answers1