given $\epsilon \gt 0$ show that:
$$\lim \limits _{n\to\infty} \frac{Vol(\{ x \in B \mid \lvert x_1 \rvert \gt \epsilon \}} {Vol(B)} \to 0$$
where $B$ is the unit ball in $n$ dimensions: $ B = \{ x \in \Bbb{R^n} \mid \lvert x \rvert \le 1 \}$.
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1$\begingroup$ Is $|x_1|> \epsilon $ as in the body or $|x_1 | < \epsilon $ as in the title? $\endgroup$– HetebrijCommented Jan 2, 2016 at 18:05
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$\begingroup$ as in the body, I edited the title $\endgroup$– Daniel KatzanCommented Jan 2, 2016 at 18:21
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1 Answer
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Hint.
You have $$\mathrm{Vol}(B_n)= \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}$$ and $$\mathrm{Vol}(\{ x \in B_n \mid \lvert x_1 \rvert > \epsilon\}) \le 2(1-\epsilon) \mathrm{Vol}(B_{n-1}) = 2 (1-\epsilon) \frac{\pi^{(n-1)/2}}{\Gamma(\frac{n-1}{2}+1)}$$
answered Jan 2, 2016 at 18:18