Timeline for The "interior" of a convex set?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2022 at 9:25 | comment | added | Yann Dubois | if that was for all b (rather than exist) that would be the definition of relative interior for convex set A. | |
| May 5, 2021 at 14:56 | comment | added | Jochen Wengenroth | In Köthe's book Topological Vector Spaces there is a notion of internal points of a convex set $A$ which means that $x$ is an interior point of the intersection of $A$ with every real line through $x$ (of course, interior refers to the usual topology of the reals). | |
| May 5, 2021 at 8:11 | comment | added | mlk | The weird thing about this definition is that it is non-local. As the cone shows, if I know $A$ only in a neighborhood of $a$, then there are cases where I cannot decide if $a\in ci(A)$ or not. Possibly a more natural condition would be "For all $b\notin A$, there exists $\epsilon$", but I think that should be equivalent to the topological interior. Maybe the important question is, what are you trying to achieve with this definition? | |
| May 5, 2021 at 5:37 | comment | added | user130903 | I see the following cases: 1. A is an affine subspace, in which case there are no convex interior points. 2. A is a pointed cone in which case the cusp is the only ci point. 3. All other cases, in which every point is ci. | |
| May 5, 2021 at 4:33 | history | edited | Lemma1 | CC BY-SA 4.0 |
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| May 5, 2021 at 4:33 | comment | added | Lemma1 | I am mainly asking if this definition is well known or equivalent to some well-known definitions. | |
| May 5, 2021 at 4:17 | review | Close votes | |||
| May 10, 2021 at 3:01 | |||||
| May 5, 2021 at 3:59 | comment | added | Ryan Budney | I'm not certain what your question is. It seems that if you write down a condition for "interior" you will have answered your own question. But if you have no criterion for interior, your question can not be answered. | |
| May 5, 2021 at 3:51 | history | edited | Lemma1 |
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| May 5, 2021 at 3:42 | history | asked | Lemma1 | CC BY-SA 4.0 |