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I am currently reading through Rockafellar's "Convex Analysis" and I am trying to make sense of Theorem 6.5:

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I understand most of the proof except for why the assumption of a finite index set is required to prove the second statement in the theorem. Rockafellar goes on to give the following example of why this is needed:

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However I am struggling to understand why the two relative interiors are not the same in his counter example. I am concerned I am missing something fundamental regarding my understanding relative interiors.

I have left the other theorems used in the proof below.

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Nick Bishop
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1 Answers1

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Note that $\text{ri}[0,1+\alpha] = (0,1+\alpha)$ and intersection of $\text{ri}[0,1+\alpha]$ over all $\alpha>0$ would be $(0,1]$. However, note that $\text{ri}[0,1] = (0,1)$. Since, the later case does not include the point $1$, the equivalence fails to hold.

Mahesh Chandra Mukkamala
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    Ah, so this finite assumption is necessary here to prevent these limiting cases, and to ensure that a prolonged line segment exists in the intersection. Thanks! – Nick Bishop Nov 24 '19 at 22:21