Intersection of tangent cones

Question

I have no Idea how to solve this question:

Let $C \subset \mathbb{R}^{n}$ be convex and closed. Prove:

$C \; is \; bounded \Leftrightarrow \cap_{x \in C} T(C,x)= \{0 \}$.

$T(C,x)$ ist the tagent cone of C in x and is defined by:

$T(C,x)=cl(\mathbb{R}_{+}(C-x))$ ,

where $cl(X)$ denotes the Closure of the set $X$.

Can you please give me an idea how to solve this? Thank you.

I tried to solve this by the definitions, but I didn't come to any constructive result. — silcrystal, Jun 23 '17 at 11:39
What you call here Tangent cone is not really Tangent cone ! — Red shoes, Jun 23 '17 at 23:45

score 0 · Answer 1 · answered Jun 23 '17 at 21:25

Hint : Show $$C \; is \; unbounded \Leftrightarrow \cap_{x \in C} T(C,x) \neq \{0 \}$$

$C$ is unbounded if and only if there is $d \neq 0 $ such that for all $x \in C $ and $\lambda \geq 0, \quad $ $x+ \lambda d \in C$

for proof

1 Answers1