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Let $C$ be a closed convex set in $\mathbb R^2$. For any $x \in C$, define $$C_x = \{y \in \mathbb R^2 \mid x + ty \in C, \forall t \ge 0\}$$ Prove that for any two points $x, x' \in C$, we have $C_x = C_{x'}$

The only thing I have been able to prove is that $0\in C_x\ \ \forall x\in C$.

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    $\begingroup$ Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ – Martin R Sep 16 at 17:42
  • $\begingroup$ First of all, does the question make sense to you? Do you see what's going on geometrically? $\endgroup$ – Théophile Sep 16 at 18:49
  • $\begingroup$ @Théophile First I thought that $x+ty$ is the line joining $x$ and $y$ but later I realized that it is incorrect. $\endgroup$ – Tabludif Sep 17 at 7:28

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