Timeline for Ellipsoids and their defining inner product
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2020 at 12:54 | vote | accept | Pavel | ||
| Dec 5, 2019 at 14:37 | answer | added | Ben McKay | timeline score: 1 | |
| Dec 5, 2019 at 14:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 5, 2019 at 13:56 | comment | added | Ben McKay | The solution is in papers of Matveev, with explicit formulas, I am pretty sure. He calls it Cauchy--Binet, or something like that. | |
| Nov 4, 2019 at 23:25 | review | Close votes | |||
| Nov 12, 2019 at 3:05 | |||||
| Nov 4, 2019 at 22:46 | answer | added | Pavel | timeline score: 1 | |
| Nov 4, 2019 at 22:39 | comment | added | Pavel | And using the same computation, the inner product g such that E=E(g) is unique. | |
| Nov 4, 2019 at 22:32 | comment | added | Pavel | Ah, I am sorry, one can just write $|T(x)|_2=|T(\frac{x}{|x|_1})|_2 |x|_1=|x|_1$ and see that any linear $T$ with $T(E_1)=E_2$ satisfies $g_2(T.,T.)=g_1(.,.)$. | |
| Nov 4, 2019 at 9:59 | comment | added | Pavel | I would still be interested in whether one can relax the orthogonality of $T$ to being symplectic or just volume-preserving. I suspect that the first case would work and the second not... | |
| Nov 4, 2019 at 9:55 | comment | added | Pavel | Great, thanks! And by expressing the Minkowski functional as $|x|_{E}=\frac{|x|}{|E\cap\langle x\rangle^+|}$, where $|.|$ is the Euclidean metric and $\langle.\rangle^+$ the positive span, it is easy to see that an orthogonal map $T$ with $T(E_1)=E_2$ preserves the Minkowski functionals, and hence $g_2(T.,T.)=g_1(.,.)$ for the associated inner products. | |
| Nov 4, 2019 at 2:40 | comment | added | user131781 | If $E$ is an ellipsoid, then its Minkowski functional is a norm on the underlying space. Furthermore it satisfies the parallelogram law and so there is a standard way to use it to rediscover the inner product. | |
| Nov 3, 2019 at 21:56 | history | asked | Pavel | CC BY-SA 4.0 |