Timeline for Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2023 at 23:29 | answer | added | M. Winter | timeline score: 2 | |
| Jan 7, 2021 at 13:55 | vote | accept | M. Winter | ||
| S Jan 7, 2021 at 0:07 | history | bounty ended | M. Winter | ||
| S Jan 7, 2021 at 0:07 | history | notice removed | M. Winter | ||
| Jan 7, 2021 at 0:07 | vote | accept | M. Winter | ||
| Jan 7, 2021 at 13:55 | |||||
| Jan 6, 2021 at 14:43 | answer | added | Guillaume Aubrun | timeline score: 7 | |
| Dec 31, 2020 at 16:25 | comment | added | David E Speyer | I think that, with the standard inner product, the actual statement should be that $\sqrt[4]{2} P$ and $\tfrac{1}{\sqrt[4]{2}} Q$ are dual; the isomorphism is still given by multiplication by $\tfrac{1+i}{\sqrt{2}}$. | |
| Dec 31, 2020 at 14:08 | comment | added | David E Speyer | @FedorPetrov Let $P$ be the convex hull of the $24$ units in the ring of half integer quaternions. Let $Q$ be the convex hull of the $24$ integer quaternions with norm $2$. Then $P$ and $\tfrac{1}{\sqrt{2}} Q$ are dual polytopes, each of which is isomorphic to the $24$-cell. Multiplication (either left or right) by $\tfrac{1+i}{\sqrt{2}}$ is an isomorphism from $P$ to $\tfrac{1}{\sqrt{2}} Q$. | |
| Dec 31, 2020 at 13:36 | comment | added | Fedor Petrov | I do not doubt, but it could (theoretically) somehow suggest other examples. | |
| Dec 31, 2020 at 13:29 | comment | added | M. Winter | @FedorPetrov I guess I could compute the matrix explicitly. This would take me some time. Why do you want to know? Do you doubt its existence? | |
| Dec 31, 2020 at 13:26 | comment | added | Fedor Petrov | what is $X$ for a 24-cell? | |
| Dec 31, 2020 at 13:19 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Dec 31, 2020 at 12:29 | comment | added | M. Winter | @FedorPetrov Yes. The paper mainly deals with the case $P^\circ=-P$ (then $P$ cannot be centrally symmetric). The two mentioned constructions, pyramids and joins, do not yield centrally symmetric polytopes either. The add-and-cut construction works only if we already have a self-polar polytope in dimension $d>4$. | |
| Dec 31, 2020 at 12:25 | comment | added | Fedor Petrov | Had you check this text? It does not seem to be about centrally symmetrical polytopes, but about self-dual. arxiv.org/pdf/1902.00784 | |
| Dec 31, 2020 at 12:00 | history | edited | M. Winter | CC BY-SA 4.0 |
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| S Dec 31, 2020 at 11:59 | history | bounty started | M. Winter | ||
| S Dec 31, 2020 at 11:59 | history | notice added | M. Winter | Draw attention | |
| Dec 23, 2020 at 10:57 | history | asked | M. Winter | CC BY-SA 4.0 |