Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$ \operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}. $$
(it is a cone in the sense that if $x \in \operatorname{cone}{C}$ and $\lambda >0$ then $\lambda x \in \operatorname{cone}{C}$).
When trying to prove Proposition 6.16 of the book "Convex Analysis and Monotone Operator Theory in Hilbert Spaces" written by Professors Heinz Bauschke and Patrick Combettes, I think that the following property of conical hull maybe true:
Let $C$ be a nonempty convex subset of $H$. Assume that $\mathrm{int}{C}\neq \varnothing$ and $0 \in C$. If $k>0$, then there exists $m>0$ such that $$ B(0,k) \cap \operatorname{cone}{C} \subset mC. $$
I guess this property by drawing some pictures of conical hull. Unfortunately, I am struggling to prove it.
Here is my question: Is the above property correct? If so, could you please give me some clues?
Any help would be appreciated.
Thank you very much for your help.
No, this is even fails in dimension $2$. Just try $C = \{(x,y) \in \mathbb R^2 \mid (x-1)^2 + y^2 \le 2 \}$.