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I am trying to prove that aff $ C \subset $ aff ri $C$ for a convex set $C$ (i.e., affine hull of the set $C$ is a subset of the affine hull of its relative interior). My question is, does $x\in C$ but $x\notin $ ri $C$ imply that $x$ is a boundary point of $C$? Also, does it matter if $C$ is convex or not?

Teodorism
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    If $x$ were not a boundary point of $C$, then $x$ would belong to the interior of $C$, so certainly $x$ would belong to the relative interior of $C$. – littleO Jul 06 '19 at 04:35

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Yes. If $x \in C$, but $x \notin \operatorname{ri} C$, then $x$ is in the boundary of $C$. Prove the contrapositive: if $x \in C$ is not in the boundary of $C$, then $x \in \operatorname{int} C$. This means $C$ contains a ball, so the affine hull of $C$ is the full space. In this case, $\operatorname{int} C = \operatorname{ri} C$. Hence, $x \in \operatorname{ri} C$.

The point of the relative interior is that boundary and interior aren't descriptive enough for general convex sets, or indeed, other (non-empty) sets. As soon as the affine hull of a non-empty set $C$ is not the full space, then $\operatorname{int} C = \emptyset$ and the boundary of $C$ covers $C$.

(Also, you should specify the space you're working in. In a general normed linear space, the result you're trying to prove is false.)

Theo Bendit
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  • Thanks. Does the point you made in your second paragraph always hold true? ("As soon as your convex set is not of full dimension, then your interior is empty and your boundary covers your whole set.") This is what I thought as well but wasn't 100% sure about it. – Teodorism Jul 06 '19 at 04:35
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    @Teodorism It is always true that, if the affine hull of your convex set is not the whole space, then the boundary covers the whole convex set. "Full dimension" doesn't really make sense outside finite dimensions. – Theo Bendit Jul 06 '19 at 04:37
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    Thanks. Does the assumption of convexity of $C$ matter? – Teodorism Jul 06 '19 at 04:37
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    @Teodorism Good point; no it does not matter. This is true for any non-empty set. – Theo Bendit Jul 06 '19 at 04:39
  • Could you please add that last point to your answer above? :) – Teodorism Jul 06 '19 at 04:40
  • @Teodorism Done. I should point out that affine hulls are really most useful with convex sets. Relative interiors are a poor topological construction without additional convex structure. – Theo Bendit Jul 06 '19 at 04:43