If $x, x' \in C$ then prove that $C_x =C_{x'}$ where C is a closed convex set

Asked Sep 06 '21 at 14:21

Active Sep 06 '21 at 15:27

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This question was part of my analysis quiz( now over) and I was unable to solve it. So, I am posting it here.

Let C be a closed convex set in $\mathbb{R}^2$.For any $x\in C$, define $C_x =$ {$y \in \mathbb{R}^2 | x+ty \in C$ for all $t\geq 0$}.

Prove that for any two points $x, x' \in C$ we have $C_x = C_{x'}$

The problem is that I am not able to think which result I should use and I am completely struck on the problem. I think it even after the quiz but still couldn't make any progress.

I have been following Tom M Apostol. Can you please give some hints?

edited Sep 06 '21 at 14:56

Martin R

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asked Sep 06 '21 at 14:21

1

Hint: For every $y \in C_x$ consider the set $S := {sx+(1-s)x'+ty \in \mathbb{R}^2: s \in (0,1), t \geq 0}.$ Is $S \subseteq C$? – bs_math Sep 06 '21 at 15:28
General hint: draw figures for your own sake! – Milten Sep 06 '21 at 19:16

If $x, x' \in C$ then prove that $C_x =C_{x'}$ where C is a closed convex set

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