Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x) \geq 0$ forall $x$ and $\displaystyle \int_{-\infty}^{\infty} f(x)\, {\rm d}x=1$. For $r \geq 0$ , define:
$$I_n(r)= \idotsint \limits_{x_1^2+x_2^2+\cdots+x_n^2 \leq r} f(x_1) f(x_2) \cdots f(x_n) \;{\rm d}(x_1, x_2, \dots, x_n)$$
Evaluate the limit $\lim I_n(r)$ for a fixed $r$.
Solution
Denote $M=\max \limits_{x\in\mathbb{R}} f(x) $ (it is finite because $f\to 0$ when $x\to \pm\infty$ ). We know that volume of $n$-dimensional ball of radius $r$ is equal to $\dfrac{\pi^{\frac{n}{2}} r^n}{ \Gamma (\frac{n}{2}+1) }$. Now, using obvious inequality and striling formula we find that:
$$I_n(r) \leq M^n \cdot \frac{\pi^{\frac{n}{2}} r^n}{ \Gamma (\frac{n}{2}+1) } \sim \frac{M^n \pi^{\frac{n}{2}} r^n}{ \sqrt{\pi n} \left( \frac{n}{2e} \right)^{n/2} } \to 0 $$
$$I_n(r)= \idotsint \limits_{x_1^2+x_2^2+\cdots+x_n^2 \leq r} f(x_1) f(x_2) \cdots f(x_n) \;{\rm d}(x_1, x_2, \dots, x_n)$$
Evaluate the limit $\lim I_n(r)$ for a fixed $r$.
Solution
Denote $M=\max \limits_{x\in\mathbb{R}} f(x) $ (it is finite because $f\to 0$ when $x\to \pm\infty$ ). We know that volume of $n$-dimensional ball of radius $r$ is equal to $\dfrac{\pi^{\frac{n}{2}} r^n}{ \Gamma (\frac{n}{2}+1) }$. Now, using obvious inequality and striling formula we find that:
$$I_n(r) \leq M^n \cdot \frac{\pi^{\frac{n}{2}} r^n}{ \Gamma (\frac{n}{2}+1) } \sim \frac{M^n \pi^{\frac{n}{2}} r^n}{ \sqrt{\pi n} \left( \frac{n}{2e} \right)^{n/2} } \to 0 $$
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