Geometry of finitely generated convex sets After drawing some pictures, I guess the following is true: Let $y_{1},\ldots, y_{d}$ be points in a Euclidean space $\mathbb{R}^{N}$ (the Euclidean norm is denoted by $\left\Vert \, \cdot \, \right\Vert$) and let $C = {\operatorname{conv}}{\left\{y_{1},\ldots,y_{d} \right\}}$ be the convex hull of those points. Assume that $0 \in C$. Then there is $r &gt;0$ such that $$\left\{ \lambda x : x \in C \text{ and } \lambda &gt;0 \text{ such that } \left\Vert \lambda x \right\Vert \leq r \right\} \subset C.$$

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.
Given that $0\in C$, you can use convexity to show that $\{ \lambda x : x \in C \text{ and } \lambda \in [0,1]\} \subset C$. From here just take $r=\min\{|x|:x\in C\}$.
Given that $0\in C$, you can use convexity to show that $\{ \lambda x : x \in C \text{ and } \lambda \in [0,1]\} \subset C$. From here just take $r=\min\{|x|:x\in C\}$.