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Consider a closed bounded convex set in the space of Lebesgue integrable functions L^P that contains the origin. Is a convex cone generated by the set closed?

Alexey Kushnir
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  • The compactness is still need to be established in my case. Hence, I would prefer to have an answer without it. However, if there is a clear positive answer with compactness, I will try to establish compactness. – Alexey Kushnir Mar 12 '21 at 18:35
  • Could you give a reference to the "trivial" part? – Alexey Kushnir Mar 12 '21 at 22:43
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    ah sorry. It is not true, if it contains the origin, even assuming compactness. Cf https://en.wikipedia.org/wiki/Conical_combination – user251257 Mar 13 '21 at 00:55
  • Thank you! I somehow missed the Wiki example. This is very helpful. I see the need for a separate specific proof for the closedness property in my case. – – Alexey Kushnir Mar 14 '21 at 12:02

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