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.
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 $ $x+ \lambda d \in C$
See If points in a convex set $C$ escape to infinity roughly in direction $v$ then an infinite ray in that direction exists
for proof