Timeline for "Paradoxes" in $\mathbb{R}^n$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 11, 2014 at 3:07 | comment | added | Nik Weaver | ... follow a greedy algorithm: choose unit vectors $v_1, v_2, \ldots$ such that $|\langle v_i,v_j\rangle| < \frac{3}{2}\epsilon$ for all $i \neq j$. By the above, as long as the balls of radius $1 - \epsilon^2$ about $\pm \epsilon v_j$ for $j < i$ don't cover the entire unit ball, a possible choice of $v_i$ exists. Since volumes scale like ${\rm radius}^n$, you need at least $\frac{1}{2}(1-\epsilon^2)^{-n}$ vectors before no choice is possible. | |
| Mar 11, 2014 at 3:04 | comment | added | Nik Weaver | Yeah, let $v$ be a unit vector and suppose $w$ is any vector in the unit ball minus the balls of radius $1-\epsilon^2$ about $\epsilon v$ and $-\epsilon v$. Then $w/\|w\|$ is a unit vector whose inner product with $v$ is at most $\frac{3}{2}\epsilon$ (to first order in $\epsilon$) in absolute value. So ... | |
| Mar 11, 2014 at 0:34 | comment | added | Campello | oops. sorry, forget that. I skipped the "unit" vectors part in the answer. Can you give a light in the "simple volume calulation"? | |
| Mar 10, 2014 at 19:09 | comment | added | Nik Weaver | Those don't look like unit vectors ... | |
| Mar 10, 2014 at 15:09 | comment | added | Campello | or, take the vectors $v_i$ to be $(\pm 10^{-5}/\sqrt{n}, \ldots, \pm 10^{-5}/\sqrt{n})$. | |
| Mar 10, 2014 at 6:29 | history | edited | Nik Weaver | CC BY-SA 3.0 |
added 5 characters in body
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| Mar 10, 2014 at 3:42 | history | answered | Nik Weaver | CC BY-SA 3.0 |