Math Problems Vault

Problem VIII.6.1

(a)(b)(c)

Prove that

?

(a)

$\frac{1}{\alpha_{n}} \int_{B^{n}} x_{1}^{2} d x=\frac{1}{n+2}$ ;

(b)

$\frac{1}{\alpha_{n}} \int_{B^{n}}\left|x_{n}\right| d x \underset {n \rightarrow \infty }{\sim } \sqrt{\frac{2}{\pi n}}$ ;

(c)

$\frac{1}{\alpha_{n}} \int_{B^{n}}\left|x_{1}\right|^{p} d x \underset {n \rightarrow \infty }{\sim } \frac{\Gamma \left(\frac{1+p}{2}\right)}{\sqrt{\pi }}\left(\frac{2}{n}\right)^{p / 2} \quad (p>-1)$ .

Problem VIII.6.2

(a)(b)

The thickness of a watermelon's rind equals $\frac{1}{20}$ of the radius of the watermelon (regarded as a ball). What percentage of the volume of the watermelon does the rind constitute

?

(a)

in three-dimensional space?

(b)

in 100-dimensional space?

Problem VIII.6.3

Prove that for any prescribed radius $R$ , in the cube $\left[0,1\right]^{n}$ of sufficiently large dimension $n$ there is a three-dimensional section containing a three-dimensional ball of radius $R$ . Thus, the terrestrial globe can be placed in an " $n$ -dimensional cubic meter."

?

Problem VIII.6.4

(a)(b)

?

(a)

The ball $B^{n}(r)$ is tangent to all edges of the $n$ -dimensional cube $\left[-1,1\right]^{n}$ . Compare, for large $n$ , the volume of the cube and the volume of the ball.

(b)

The cube $\left[-1,1\right]^{n}$ is divided by the coordinate hyperplanes into $2^{n}$ cubes. In each of them we inscribe a ball, and then consider the ball $B^{n}(\rho )$ that is externally tangent to all the balls inscribed in the cubes. For which $n$ does the inclusion $B^{n}(\rho ) \subset [-1,1]^{n}$ hold? Which is larger for large $n$ : the volume of the cube $\left[-1,1\right]^{n}$ or the volume of the ball $B^{n}(\rho )$ ?

Problem VIII.6.5

(a)(b)(c)

Let $I=\left[-\frac{1}{2}, \frac{1}{2}\right], \quad r \in \mathbb {R}, \quad \omega \in \mathbb {R}^{n}, \quad \omega \neq 0$ . Further, let $H_{n-1}(\omega , r)=\left\{ x \in I^{n} \mid (x, \omega )=r\right\}$ be the cross-section of the cube $I^{n}$ by the hyperplane $(x, \omega )=r$ .

?

(a)

Prove that

\lambda _{n-1}\left(H_{n-1}(\omega , r)\right)=\frac{2}{\pi }\left\| \omega \right\| \int _{0}^{+\infty } \cos (2 r t) \prod _{j=1}^{n} \frac{\sin \left(\omega _{j} t\right)}{\omega _{j} t} d t

(b)

Find the limit as $n \rightarrow +\infty$ of the measure of the central cross-section orthogonal to the main diagonal of the cube, i.e., the limit of the quantity $\lambda_{n-1}\left(H_{n-1}((1, \ldots , 1), 0)\right)$ .

(c)

Prove that $\lambda_{n-1}\left(H_{n-1}(\omega , r)\right) \leqslant \sqrt{2}$ for all $n=2,3, \ldots$ , $r \in \mathbb {R}$ and $\omega \in \mathbb {R}^{n}, \omega \neq 0$ . For which cross-sections does equality hold?

Problem VIII.6.6

Alice, having found herself in a 1000-dimensional space, was asked how to make thin glass hoops (spherical belts) so that their width equals $\frac{1}{10}$ of the diameter. "Nothing could be simpler," Alice replied, "one must blow spheres, and then cut off the excess" (Fig. 13). What will be the percentage of waste

Fig. 13 Fig. 13

with this technology?

?

Problem VIII.6.7

Let $a \in \mathbb {R}^{n},\left\| a\right\| =1$ . A point $x$ is chosen at random in the ball $B^{n}$ . Which is more probable for large $n:\left|(a, x)\right|<10^{-3}$ or $\left|(a, x)\right|>10^{-3}$ ?

?

Problem VIII.6.8

Two points are chosen at random on the sphere $S^{n-1}$ . Let $L_{n}$ be the mean value of the distance between them. Prove that $\lim L_{n} \rightarrow \sqrt{2}$ . This result shows that for large $n$ the radii drawn to two randomly chosen points on the sphere are, "as a rule," almost orthogonal.

?

Problem VIII.6.9

Let $P_{n}$ be the normalized Lebesgue measure on the sphere $S^{n-1}(\sqrt{n})$ . Prove that the coordinates on this sphere are "asymptotically normally distributed," i.e., for any $a, b \in \mathbb {R}, a<b$ , the relation holds

P_{n}\left\{ \left(x_{1}, \ldots , x_{n}\right) \in S^{n-1}(\sqrt{n}) \mid a<x_{n}<b\right\} \underset {n \rightarrow \infty }{\longrightarrow } \frac{1}{\sqrt{2 \pi }} \int _{a}^{b} e^{-\frac{t^{2}}{2}} d t .

?

Problem VIII.6.10

Let the function $f$ be continuous on the closed ball $\vec{B}^{n}$ , $M=\max_{\overline{B^{n}}}\left|f\right|$ , and suppose that for any $\omega \in S^{n-1}$ and $r \in [0,1]$ the estimate $\left|f(r \omega )-f(\omega )\right| \leqslant h(1-r)$ holds, where $h$ is an increasing function, continuous on the interval $\left[0,1\right]$ , vanishing at zero. Prove that the mean value of the function $f$ on the ball $B^{n}$ and on the sphere $S^{n-1}$ are close in the following sense: for any $\varepsilon \in \left(0, \frac{1}{2}\right)$

\left|\frac{1}{\alpha _{n}} \int _{B^{n}} f(x) d x-\frac{1}{\sigma _{n}} \int _{S^{n-1}} f(\omega ) d \mu _{n-1}(\omega )\right| \leqslant 2\left(M(1-\varepsilon )^{n}+h(\varepsilon )\right) .

?

Problem VIII.6.11

Prove that for any $q \in \mathbb {R}$ the integrals $\int_{\mathbb {R}^{n}}\left|x_{1} \ldots x_{n}\right|^{\frac{q}{n}} d \gamma_{n}(x)$ have, as $n \rightarrow \infty$ , a finite positive limit.

?

Problem VIII.6.12

Prove that for any $q \in \mathbb {R}, 0<p<\infty$

\frac{1}{\alpha _{n}} \int _{B^{n}}\left\| x\right\| _{p}^{q} d x \underset {n \rightarrow \infty }{\sim } \frac{1}{\sigma _{n-1}} \int _{S^{n-1}}\left\| x\right\| _{p}^{q} d \mu _{n-1}(x)

?

Problem VIII.6.13

(a)(b)(c)

Let $\tilde{\gamma }_{n}$ be the measure in $\mathbb {R}^{n}$ with density $\left(\frac{n}{2 \pi }\right)^{n / 2} e^{-n\left\| x\right\|^{2} / 2}$ ;

?

(a)

verify that $\widetilde{\gamma }_{n}\left(\mathbb {R}^{n}\right)=1$ ;

(b)

find the limit $\lim_{n \rightarrow \infty } \widetilde{\gamma }_{n}\left(B^{n}\right)$ ;

(c)

prove that for $0<\varepsilon <1$

\int _{|\left\| x\right\| -1|>\varepsilon } d \widetilde{\gamma }_{n}(x)<2 \sqrt{n} \cdot e^{-n \varepsilon ^{2} / 2}

Problem VIII.6.14

Prove that for any $q \in \mathbb {R}, 0<p \leqslant \infty$

\int _{\mathbb {R}^{n}}\left\| x\right\| _{p}^{q} d \tilde{\gamma }_{n}(x) \underset {n \rightarrow \infty }{\sim } \frac{1}{\sigma _{n-1}} \int _{S^{n-1}}\left\| x\right\| _{p}^{q} d \mu _{n-1}(x)

(for the definition of the measure $\tilde{\gamma }_{n}$ see problem 6.13).

?

Problem VIII.6.15

Prove that $\int_{\mathbb {R}^{n}}\left\| x\right\|^{q} d \gamma_{n}(x) \underset {n \rightarrow \infty }{\sim } n^{q / 2}(q \in \mathbb {R})$ .

?

Problem VIII.6.16

Prove that for any $q \in \mathbb {R}$

\int _{\mathbb {R}^{n}}\left\| x\right\| _{\infty }^{q} d \gamma _{n}(x) \underset {n \rightarrow \infty }{\sim }(2 \ln n)^{q / 2}

?

Problem VIII.6.17

(a)(b)

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Let $\quad q \in \mathbb {R}, \quad m, n \in \mathbb {N}, \quad 2 \leqslant m \leqslant n, \quad m+q>0, \quad x \in \mathbb {R}^{n}$ , $x=\left(x_{1}, \ldots , x_{n}\right)$ ,

\mathcal{Y}_{m, n}=\frac{1}{\sigma _{n-1}} \int _{S^{n-1}} \max _{1 \leqslant k \leqslant m}\left|x_{k}\right|^{q} d \mu _{n-1}(x)

Prove that

?

(a)

$\mathcal{Y}_{m, n} \asymp \left(\frac{\ln m}{n}\right)^{q / 2}$ ;

(b)

$\mathcal{Y}_{m, n} \underset {m \rightarrow \infty }{\sim }\left(\frac{2 \ln m}{n}\right)^{q / 2}$ .

Problem VIII.6.18

(a)(b)(c)(d)

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Prove that for any $q \in \mathbb {R}, 0<p<\infty$

\mathcal{Y}_{n}(p, q)=\int _{\mathbb {R}^{n}}\left\| x\right\| _{p}^{q} d \gamma _{n}(x) \asymp n^{q / p} .

Consider successively the cases

?

(a)

$q>0, \quad 0<p \leqslant 2$ ;

(b)

$0<q \leqslant p, \quad p \geqslant 2$ ;

(c)

$2 \leqslant p \leqslant q$ ;

(d)

$q<0$ .

Problem VIII.6.19

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Prove that for any $q \in \mathbb {R}, 0<p<\infty$

\frac{1}{\sigma _{n-1}} \int _{S^{n-1}}\left\| x\right\| _{p}^{q} d \mu _{n-1}(x) \asymp n^{\frac{q}{p}-\frac{q}{2}}

?

Problem VIII.6.20

(a)(b)

?

(a)

Prove that the radius of the ball $B^{n}\left(r_{n}\right)$ whose volume equals one is expressed by the formula

r_{n}=\sqrt{\frac{n}{2 \pi e}}\left(1+a \frac{\ln n}{n}+\frac{b}{n}+o\left(\frac{1}{n}\right)\right),

and find the coefficients $a$ and $b$ . What does the Lebesgue measure (of dimension $n-1$ ) of the section of this ball by a hyperplane passing through its center tend to?

(b)

Find the asymptotics of the radius $\rho_{n}$ of the ball $B^{n}\left(\rho_{n}\right)$ whose Gaussian volume equals $\frac{1}{2}$ .

Problem VIII.6.21

Let $A$ and $B$ be convex centrally symmetric (with center at zero) compacta in $\mathbb {R}^{n}$ . Can one assert that $\lambda_{n}(A) \leqslant \lambda_{n}(B)$ , if every central section of the set $A$ by a hyperplane has Lebesgue measure (of dimension $n-1$ ) smaller than the corresponding section of the set $B$ ? (Compare, for large $n$ , the sections of the unit cube and of the ball having unit volume.)

?

Problem VIII.6.22

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Let $a_{n}, b_{n}, c_{n}$ be the average values of the function $\frac{1}{\left\| x\right\|_{1}}$ on the sets $O_{n}, B^{n},\left[-1,1\right]^{n}$ , where $O_{n}$ is the $n$ -dimensional octahedron: $O_{n}=\left\{ \left(x_{1}, \ldots , x_{n}\right) \mid \sum_{1 \leqslant k \leqslant n}\left|x_{k}\right| \leqslant 1\right\}$ . Prove that

a_{n}=\frac{n}{n-1}, \quad b_{n} \asymp \frac{1}{\sqrt{n}}, \quad c_{n} \asymp \frac{1}{n} .

?

Problem VIII.6.23

(a)(b)(c)

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Let $I_{n}, J_{n}, K_{n}$ be the average values of the function $\frac{\left\| x\right\|_{2}^{2}}{\left\| x\right\|_{1}}$ on the octahedron $O_{n}$ , the ball $B^{n}$ , and the cube $\left[-1,1\right]^{n}$ respectively. Prove that

?

(a)

$I_{n} \sim \frac{2}{n}$ ;

(b)

$J_{n} \asymp \frac{1}{\sqrt{n}}$ ;

(c)

$K_{n} \asymp 1$ .

Problem VIII.6.24

(a)(b)

Let $O_{n}+\varepsilon B^{n}-\varepsilon$ be the neighborhood of the octahedron $O_{n}$ .

?

(a)

Find the limit $\lim_{n \rightarrow +\infty }\left(\frac{\lambda_{n}\left(O_{n}+\varepsilon B^{n}\right)}{\alpha_{n}}\right)^{1 / n}$ .

(b)

Find the asymptotics of the ratio $\frac{\lambda_{n}\left(O_{n}+\varepsilon B^{n}\right)}{\alpha_{n}}$ as $n \rightarrow +\infty$ .

Problem VIII.6.25

Prove that $\lambda_{n}\left(A_{n}\right) \sim 2\left(\frac{2}{\pi }\right)^{n+1}$ , where

A_{n}=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n} \mid x_{k}+x_{k+1} \leqslant 1 \quad (1 \leqslant k<n) ; \, x_{1} \geqslant 0, \ldots , x_{n} \geqslant 0\right\} .

?

Problem VIII.6.26

Let $V_{n}$ be a convex set, centrally symmetric about zero, contained in the cube $(-1,1)^{n}$ . Prove that if $\lambda_{n}\left(V_{n}\right)>\left(\frac{3}{2}\right)^{n}$ , then for sufficiently large $n$ the set $2 V_{n}$ contains a point $a=\left(a_{1}, \ldots , a_{n}\right)$ with integer coordinates such that $\sum_{1 \leqslant k \leqslant n}\left|a_{k}\right|>\frac{n}{10}$ .

?

Problem VIII.6.27

(a)(b)

Find the asymptotics of the integrals

?

(a)

$\int_{\left[\frac{1}{n}, 1\right]^{n}} \frac{\min x_{k}}{\max x_{k}} d x$ ;

(b)

$\int_{\left[\frac{1}{n}, 1\right]^{n}} \max_{\overline{\min } x_{k}} d x$ .

Problem VIII.6.28

(a)(b)(c)(d)

Prove that as $n \rightarrow +\infty$

?

(a)

$\frac{1}{\sqrt{n}} \int_{\left[0,1\right]^{n}}\left\| x\right\| d x \rightarrow \frac{1}{\sqrt{3}}$ ;

(b)

$\int_{\left[0,1\right]^{n}} f\left(\frac{x_{1}+\ldots +x_{n}}{n}\right) d x \rightarrow f\left(\frac{1}{2}\right) \quad (f \in C(\left[0,1\right]))$ ;

(c)

$\int_{\left[0,1\right]^{n}} f\left(\sqrt[n]{x_{1} \ldots x_{n}}\right) d x \rightarrow f\left(\frac{1}{e}\right) \quad (f \in C(\left[0,1\right]))$ ;

(d)

$\int_{\mathbb {R}^{n}} f\left(\frac{\left|x_{1}\right|+\ldots +\left|x_{n}\right|}{n}\right) d \gamma_{n}(x) \rightarrow f\left(\sqrt{\frac{2}{\pi }}\right)\left(f \in C\left(\mathbb {R}^{n}\right), \sup \left|f\right|<\infty \right)$ .

Problem VIII.6.29

(a)(b)(c)

Let $f \geqslant 0, \int_{-\infty }^{+\infty } f(t) d t=1,0<p<\infty$ . Prove that as $n \rightarrow +\infty \left(\right.$ hereafter $\left.x=\left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n}\right)$

?

(a)

$\quad \int_{\left\| x\right\|_{p} \leqslant c} f\left(x_{1}\right) \cdot f\left(x_{2}\right) \ldots f\left(x_{n}\right) d x \rightarrow 0$ for any $c>0$ ;

(b)

$\int_{\left\| x\right\|_{p} \leqslant n^{\theta }} f\left(x_{1}\right) \cdot f\left(x_{2}\right) \ldots f\left(x_{n}\right) d x \rightarrow 0$ for any $\theta \in \left(0, \frac{1}{p}\right)$ .

(c)

Can $n^{\theta }$ in part b) be replaced by $\left(n \omega_{n}\right)^{1 / p}$ , where $\omega_{n}>0, \omega_{n} \rightarrow 0$ ?

Problem VIII.6.30

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Let $p>1, f \in \mathscr {L}^{p}(0,1),\left\| f\right\|_{p} \neq 0$ . Prove that as $n \rightarrow \infty$

I_{n}=\int _{(0,1)^{n}}\left(\sum _{k=1}^{n}\left|f\left(x_{k}\right)\right|^{p}\right)^{1 / p} d x \asymp n^{1 / p}

(the coefficients of the relation $\asymp$ depend on $f$ and $p$ ).

?

Problem VIII.6.31

(a)(b)

For positive sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$ the notation $a_{n} \asymp b_{n}$ means that $0<\inf_{n} \frac{a_{n}}{b_{n}} \leqslant \sup_{n} \frac{a_{n}}{b_{n}}<\infty$ (the sequences are of the same order); the notation is used in the same sense for arbitrary families of positive numbers. Let $F$ be the decreasing distribution function of a measurable function $f$ on $(0,1)$ . As is known, if $f \in \mathscr {L}^{p}(0,1)$ , then $F(t)=o\left(1 / t^{p}\right)$ as $t \rightarrow +\infty$ (see problem 3.13). Suppose that $F$ satisfies only the weaker condition: $F(t)=O\left(1 / t^{p}\right)$ as $t \rightarrow +\infty$ . Prove that in this case, for $p>1$ ,

?

(a)

the integral $I_{n}$ from the previous problem admits the estimate

I_{n}=O\left((n \ln (n+1))^{1 / p}\right) ;

(b)

this estimate is sharp in the following sense:

\text{ if } \quad F(t) \asymp 1 / t^{p} \quad \text{ as } \quad t \rightarrow +\infty , \quad \text{ then } \quad I_{n} \asymp (n \ln (n+1))^{1 / p} .

Problem VIII.6.32

Prove that for any $n$ and any vector $a \in \mathbb {R}^{n}$ the inequality

A_{p}\left\| a\right\| \leqslant \left(2^{-n} \int _{\left[-1,1\right]^{n}}\left|(x, a)\right|^{p} d x\right)^{1 / p} \leqslant B_{p}\left\| a\right\|

holds, where $A_{p}, B_{p}$ are positive constants depending only on $p>0$ (cf. Problem 3.37 a)).

?