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Consider the set of points $S = \{(\cos(t), \sin(t))\; |\; 0\leq t \leq \pi/2\}$ (an arc describing a quarter circle)

Is it true that the relative interior of $S$ consists of all points of the arc except the end points, $(1,0)$ and $(0,1)$ ?

I am trying to show that by the definition $\mathrm{relint}(S) := \{x \in S\;| \; \exists \;\delta > 0, B_\delta(x) \cap \mathrm{aff}(S) \subseteq S\}$ where $\mathrm{aff}(S)$ denotes the affine hull of $S$.

I am stuck because $\mathrm{aff}(S)$ is all of $\mathbb{R}^2$, hence for any $x \in S$, $\delta >0, B_\delta(x) \cap \mathrm{aff}(S) = B_\delta(x)$ which is a ball in $\mathbb{R}^2$, and hence never a subset of $S$. Thus I get $\mathrm{relint}(S) = \{ \}$.

me10240
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  • See here: https://math.stackexchange.com/questions/1144750/whats-the-difference-between-interior-and-relative-interior – Christian Blatter May 29 '19 at 09:19
  • @ChristianBlatter Do you think OP misunderstands the definition of relative interior? They have given a correct proof that the relative interior of $S$ is empty. If I'm not mistaken, I think the key point for OP is just to know that their argument is correct, and that the thing they had set out to prove is indeed false. – littleO May 29 '19 at 09:37

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The relative interior of $S$ is empty, for the reason you said.

littleO
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  • In another question, that explained what rel interior means, the answer says that the interior of a line segment in R^2 is empty, but the rel interior of the line will consist of all except the end points. My conclusion here seems to contradict that answer. – me10240 May 29 '19 at 07:52
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    @me10240 Note that the affine hull of a line segment is a line, not all of $\mathbb R^2$. There is no contradiction. – littleO May 29 '19 at 07:55
  • What's the reason for the delete vote? If there's an error in what I wrote, I'd like to know. – littleO May 29 '19 at 09:09
  • That quarter circle can be equivalently written as ${x| x_1^2+x_2^2 = 1, x_1 \geq 0, x_2 \geq 0}$. The affine hull is a subset of $\mathbb{R}^2$, so that the relative interior of this arc is still with respect to the 2D balls, so it is empty. Is this correct? – Olórin Oct 13 '19 at 04:24