⧼exchistory⧽
3 exercise(s) shown, 0 hidden
BBy Bot
Jun 01'24
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Let [math]\Bbar_1(0)[/math] denote the unit ball of [math]\mathbb{R}^d[/math].
- Use Lemma to compute [math]\lambda^d(\Bbar_1(0))[/math] for [math]d=1,\dots,10[/math].
- Compute [math]\lim_{d\rightarrow\infty}\lambda^d(\Bbar_1(0))[/math].
- Show that [math]\lambda^d(\Bbar_1(0))=\medfrac{\pi^{d/2}}{\Gamma(d/2+1)}[/math] holds, where [math]\Gamma\colon(0,\infty)\rightarrow\mathbb{R}[/math] denotes the Gamma function.
BBy Bot
Jun 01'24
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[/math]
Let [math]d\geqslant3[/math] and [math]n[/math] be such that [math]2\ln(n)\leqslant d[/math] holds. Show that
[[math]]
\P\bigl[\bigl|\bigl\langle{}\medfrac{x^{\scriptscriptstyle(j)}}{\|x^{\scriptscriptstyle(j)}\|},\medfrac{x^{\scriptscriptstyle(k)}}{\|x^{\scriptscriptstyle(k)}\|}\bigr\rangle{}\bigr|\leqslant\medfrac{\sqrt{6\ln n}}{\sqrt{d-1}} \text{ for all }j\not=k\bigr]\geqslant 1-\medfrac{1}{n}
[[/math]]
holds, when [math]x^{\scriptscriptstyle(1)},\dots,x^{\scriptscriptstyle(n)}[/math] are drawn uniformly at random from the [math]d[/math]--dimensional unit ball.
Hint: Use that [math]1\leqslant d^2/(d-2\ln n)^2[/math] holds and apply then the Theorem of Total Probability.