Surface area - even dimensions

Question

Consider the surface:

$$ (\log x_1)^2+(\log x_2)^2+\cdot\cdot\cdot +(\log x_n)^2=R $$

For $R=1$ and in even dimensions $n=2,4,6, \cdot\cdot\cdot$ we have the volumes:

$$V=\bigg( \frac{\pi^2 I_1(\sqrt{2})}{\sqrt{2}}, \frac{3\pi^3 I_2(2)}{8}, \frac{5\pi^4 I_3(\sqrt{6})}{12\sqrt{6}}, \cdot\cdot\cdot \bigg) $$

Where $I_n$ is the modified Bessel function of the first kind.

To obtain these volumes we let $B_n$ denote $\{x_1^2+\ldots+x_n^2\leq 1\}$ such that the integral equals: $$\int_{B_n}\exp\sum x_i\,d\mu = \int_{B_n}\exp(\sqrt{n} x_1)\,d\mu=\int_{-1}^{1}\exp(\sqrt{n}x)\frac{\pi^{n/2}}{\Gamma(1+n/2)}(1-x^2)^{(n-1)/2}dx $$

What is the surface area for even dimensions? Is it related to the volumes?

Answer 1

I don't believe that a closed form expression for the (hyper-)area exists, e.g. the integral for $n=2$ is $$ \int_0^{2\pi}d\theta \,\sqrt{e^{2\sin\theta}\cos^2\theta+e^{2\cos\theta}\sin^2\theta}\approx 7.51018. $$ However, the volume can indeed be calculated like you suggest. You must have made a mistake in solving the integrals.

Using your change of variables $y_i=\log x_i$, we find $$ V_n=\int_{B_n}\exp \sum_i y_i\, d\mu= \frac{2\pi^{\frac{n-1}{2}}}{\Gamma(\frac{n-1}{2})}\int_0^1\int_0^\pi e^{\sqrt{n}r\cos\theta}\sin^{n-2}\theta r^{n-1}d\theta\,dr = \frac{(2\pi)^{n/2}I_{n/2}(\sqrt{n})}{n^{n/4}}. $$ Evaluating at $n=2$, yields a value in accordance with the value found by Jean Marie $$ V_2 \approx 3.99524. $$

Answer 2

Here is the study of the particular case $n=2$ where I am in disagreement with your results.

The implicit equation for this curve is

$$(\ln(x))^2+(\ln(y))^2=1\tag{1}$$

and the 2 cartesian equations are :

$$y=\exp\left(\pm \sqrt{1-\ln(x)^2}\right)\tag{2}$$

(blue curve for the + sign ; red curve for the - sign in (2))

The area, enclosed into the square $\left[\frac{1}{e},e\right] \times \left[\frac{1}{e},e\right]$ has approximate value $3.99524$ different from the value given in the text :

$$\pi^2 \frac{I_1(\sqrt{2})}{\sqrt{2}} \approx 6.2757$$

(approximate value obtained thanks to Wolfram Alpha)

How can this been explained ?

Reamrk : I find it in particular difficult to see which change of variables you have precisely used in your integrals.