Timeline for Support function of the intersection of a hyper-ellipsoid and a Euclidean ball
Current License: CC BY-SA 4.0
15 events
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| Feb 5, 2023 at 17:51 | comment | added | dohmatob | I've corrected the bugs in the statement of the question and posted a complete solution in an edit. The bugs where due to the fact that what I defined as $\omega$ was meant to be the dual norm of $\omega$ (i.e the support function of its unit ball). Thanks for the input | |
| Feb 5, 2023 at 17:47 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 5, 2023 at 17:41 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 5, 2023 at 17:32 | comment | added | Iosif Pinelis | Yes, the proof is very simple. | |
| Feb 5, 2023 at 17:30 | comment | added | dohmatob | Hum, yes I think I see how to get the proof, by simply observing that $1/\min(a,b) = \max(1/a,1/b) \le 1/a + 1/b$. This would give the $\sqrt 2$ factor. | |
| Feb 5, 2023 at 17:22 | comment | added | Iosif Pinelis | Also, your desired conclusion seems to be just stated, without proof, in the paper linked in your comment -- even though the proof is simple. | |
| Feb 5, 2023 at 17:15 | comment | added | dohmatob | Indeed, there are a couple of typographical mistakes. Editing. Thanks. | |
| Feb 5, 2023 at 17:12 | comment | added | Iosif Pinelis | Two more mistakes remaining there (try replacing $\lambda_j$ and $r^2$ by $t^2\lambda_j$ and $t^2r^2$). | |
| Feb 5, 2023 at 17:12 | comment | added | dohmatob | Concerning your, other comments, it is indeed shown in the reference that $D(r) \subseteq \mathcal E(r) \subseteq \sqrt{2} D(r)$, where $D(r) := \{x \in \mathbb R^d \mid \sum_j x_j^2/\lambda_j(r) \le 1\}$. There is no randomness involved here. The claim then follows upon comparing the support functions of these sets . No ? | |
| Feb 5, 2023 at 17:09 | comment | added | dohmatob | I omitted a square root sign in the definition of $\omega$. Fixed. | |
| Feb 5, 2023 at 17:07 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 5, 2023 at 17:03 | comment | added | Iosif Pinelis | (i) Your inequality cannot be true, as it is not homogeneous in $x$. There seem to be three mistakes in the definition of $\omega$. (ii) Theorem 2.1 in the paper linked in your comment seems rather different from whatever you wanted to ask here. In particular, it involves random variables, in contrast with your post. | |
| Feb 5, 2023 at 16:51 | comment | added | dohmatob | Sorry for the noise, but it turns our the problem is solved by an argument of Theorem 2.1 of this paper jmlr.org/papers/volume4/mendelson03a/mendelson03a.pdf. I'll close the question if nobody objects to it. | |
| Feb 5, 2023 at 16:42 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 5, 2023 at 16:31 | history | asked | dohmatob | CC BY-SA 4.0 |