Timeline for Fix positive $t$. Construct $a_n \in \mathbb R^n$ such that $(\inf_x \|x-a_n\|_2 + t\|x\|_1 )/\min(\|a_n\|_2,t\|a_n\|_1) \to 0$
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Oct 13, 2022 at 6:04 | vote | accept | dohmatob | ||
| Oct 10, 2022 at 21:29 | answer | added | Iosif Pinelis | timeline score: 3 | |
| Oct 10, 2022 at 19:52 | comment | added | Christian Remling | I don't think this is possible. Let's say $t=1$, to keep it simple, and wlog $\|a\|_2=1$, so $M=1$. To make $K$ small, we'd have to take $x=a+b$, $\|b\|_2<\epsilon$, but then $\|x\|_1\ge \|a+b\|_2\ge 1-\epsilon$. | |
| Oct 10, 2022 at 19:40 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Oct 10, 2022 at 19:15 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Oct 10, 2022 at 19:13 | comment | added | dohmatob | I added some motivation. Also, one can compute $R_1(a,t) = \min(1,t)|a|/\min(1,t)|a|=1$ for all $a \ne 0$ and positive $t$. The problem becomes interesting in the limit $n \to \infty$. | |
| Oct 10, 2022 at 19:00 | comment | added | Aryeh Kontorovich | Can you motivate this a bit? Do you know the answer for $n=1$? | |
| Oct 10, 2022 at 18:26 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Oct 10, 2022 at 18:20 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Oct 10, 2022 at 18:15 | history | asked | dohmatob | CC BY-SA 4.0 |