Timeline for Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 8, 2023 at 9:10 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Put in a proof of the critical $3$-variable inequality.
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| May 7, 2023 at 17:35 | vote | accept | Spencer Kraisler | ||
| May 8, 2023 at 16:43 | |||||
| May 6, 2023 at 18:19 | history | edited | Robert Bryant | CC BY-SA 4.0 |
A second major rewrite to clean things up.
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| May 6, 2023 at 18:14 | comment | added | Spencer Kraisler | No worries. But if it holds for unit quaternions, and the unit quaternions are a double cover of $SO(3)$, I would think there's some nifty isometry trick one can use to match everything to $SO(3)$. At the end of the day, they're both Lie groups and we're dealing with two different embeddings of the "same" manifold (not the same ofc). | |
| May 6, 2023 at 17:52 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 309 characters in body
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| May 6, 2023 at 15:14 | history | edited | Robert Bryant | CC BY-SA 4.0 |
A major revision to fix a serious error, because the 'counterexample' did not work.
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| May 6, 2023 at 15:10 | comment | added | Robert Bryant | @SpencerKraisler: I realized my error, which was applying the cosine function to reverse the inequality on a range where cosine is not strictly decreasing. That's why the cosine'd inequality fails but it doesn't give a counterexample to your inequality. Things go better for the quaternions, and you'd think that would also work for $\mathrm{SO}(3)$, but I'm checking the details. | |
| May 6, 2023 at 12:31 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Removed a spurious leftover phrase in the last parenthetical remark.
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| May 6, 2023 at 9:54 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Cleaned up the exposition a little bit.
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| May 6, 2023 at 1:02 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a remark about the quaternion case (which is different from the SO(3) case).
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| May 5, 2023 at 23:01 | comment | added | Spencer Kraisler | Hm, if you provide me numerical evidence of this counterexample, I will select your answer. However, from my script, this example does not flip my inequality. | |
| May 5, 2023 at 16:08 | comment | added | Spencer Kraisler | Hm, I plugged this in matlab and the inequality still holds for me. I, in particular, set $x=2.5$. Although my code could be incorrect, I highly doubt it considering my script is simple (just matrix exponential and logarithms, and traces). I'm happy to share my code with you. | |
| May 5, 2023 at 14:24 | history | answered | Robert Bryant | CC BY-SA 4.0 |