Timeline for The set of strongly positive forms is a closed cone
Current License: CC BY-SA 4.0
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| Sep 8 at 0:42 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 7 at 2:35 | vote | accept | Junyu Cao | ||
| Sep 7 at 2:35 | comment | added | Junyu Cao | Great, it's now proved! And I think you should mention the proposition: "If S is a non-empty convex compact set which does not contain the origin, then the convex conical hull of S is a closed set." Since it is non-trivial for someone who first meets this. | |
| Sep 7 at 2:18 | comment | added | Skywalker | You are right. But in our case the counter example will not happen. I have provide the detail of my proof. | |
| Sep 7 at 2:17 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 7 at 0:17 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 7 at 0:08 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 6 at 23:56 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 6 at 22:32 | comment | added | Junyu Cao | add a counter-example here, which shows that the convex hull of a closed cone (in your definition) is not closed. math.stackexchange.com/questions/2791730/…, you can check it | |
| Sep 6 at 22:27 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 6 at 22:21 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 6 at 22:10 | history | edited | Skywalker | CC BY-SA 4.0 |
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| Sep 6 at 21:45 | comment | added | Skywalker | It is easy to prove that the convex hull of a closed cone(not just a closed set) is closed. You can try it by yourself. And my definition of cone is that if x is in C, then for a>0, ax is also in C. I don't require it to be convex. So it should be fine that the set of simple forms is a cone. | |
| Sep 6 at 20:57 | comment | added | Junyu Cao | And just a small typo: the set of simple strongly positive forms is a closed set (shown in your answer), but NOT a cone (since the sum of two simple strongly positive forms is not necessarily simple). | |
| Sep 6 at 20:49 | comment | added | Junyu Cao | The use of Plücker's embedding here is great. However, the convex hull of a closed set is not always closed, as pointed in math.stackexchange.com/questions/340324/…. So I think the proof is not finished yet. | |
| Sep 6 at 20:38 | vote | accept | Junyu Cao | ||
| Sep 6 at 20:45 | |||||
| Sep 6 at 20:13 | history | answered | Skywalker | CC BY-SA 4.0 |