Timeline for If $|P|<\infty$ and $C=P\cap\partial(\textrm{Conv}(P))$, then $P\subset\textrm{Conv}(C)$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 9, 2023 at 13:52 | comment | added | Iosif Pinelis | @one-day-at-a-time : You are welcome. These MathOverflow guidelines may be relevant in this case. | |
| Nov 9, 2023 at 10:08 | comment | added | one-day-at-a-time | Everything is clear. Sorry I am a newbie, so I just realized that I have to accept the answer. Thank you again, sir. | |
| Nov 9, 2023 at 10:06 | vote | accept | one-day-at-a-time | ||
| Nov 6, 2023 at 16:06 | comment | added | one-day-at-a-time | Thank you very much for your kind guide. :D | |
| Nov 6, 2023 at 15:31 | comment | added | Iosif Pinelis | @one-day-at-a-time : Since your set $P$ is finite, without loss of generality it is a subset of $\mathbb R^d$ for some $d$. The best book on this topic is Convex Analysis by R. Tyrrell Rockafellar. I will refer to the 1970 edition of this book. $K=\conv\ext K$ indeed follows from the Krein--Milman theorem, or from Corollary 18.5.1 of Rockafellar's book. $\ext K=P\cap\ext K$ or, equivalently, $\ext K\subseteq P$ follows from Corollary 18.3.1 of the same book. $\ext K\subseteq\partial K$ is an easy exercise, following immediately from the definitions of $\ext K$ and $\partial K$. | |
| Nov 6, 2023 at 9:18 | comment | added | one-day-at-a-time | It seems $C$ is also referred to as vertices. Since I am not familiar with this field, I am looking for a guide. | |
| Nov 6, 2023 at 8:58 | comment | added | one-day-at-a-time | I think the first one is from the Krein-Milman theorem. | |
| Nov 6, 2023 at 8:48 | comment | added | one-day-at-a-time | Thank you very much. Is there any reference (books, lecture notes, or books) where I can study proofs of $K=\textrm{conv } \textrm{ext } K$, $\textrm{ext } K = P\cap\textrm{ext } K$, and $\textrm{ext }K\subseteq\partial K$? | |
| Nov 5, 2023 at 15:50 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 84 characters in body
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| Nov 5, 2023 at 15:43 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |