This question is a generalization of the question [Volume ratio of $\ell\_1$ balls and $\ell\_1$ surfaces](https://mathoverflow.net/questions/281611/volume-ratio-of-ell-1-balls-and-ell-1-surfaces)


For any $p\in[1,\infty]$ define $\|x\|\_p := (|x\_1|^p+\cdots+|x\_d|^p)^{1/p}$ for $p\in[1,\infty)$ and $\|x\|\_\infty := \max\_{1\leq i\leq d}|x\_i|$ for $p=\infty$. Denote $B\_p^d := \{x\in\mathbb R^d: \|x\|\_p\leq 1\}$ as the unit $\ell\_p$-ball in $d$ dimension, and let $\partial B\_p^d := \{x\in\mathbb R^d: \|x\|\_p = 1\}$. The quantity of interest is the following "ratio"
$$
\mathfrak d\_{p,d} := \frac{\mathrm{vol}\_{d-1}(\partial B\_p^d)}{\mathrm{vol}\_{d}(B\_p^d)}.
$$


My question is the following:



> 
> For any fixed $p\in[1,\infty]$, how does $\mathfrak d\_{p,d}$ asymptotically scale with dimension $d$, as $d\to\infty$?
> 
> 
> 


For some special cases, both $\mathrm{vol}\_d(B\_p^d)$ and $\mathrm{vol}\_{d-1}(\partial B\_p^d)$ have closed form solutions and the ratio $\mathfrak d\_{p,d}$ can be calculated explicitly. Below are three examples:


1. $p=2$: in this case $\mathrm{vol}\_d(B\_2^d) = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$, $\mathrm{vol}\_{d-1}(\partial B\_2^d) = \frac{2\pi^{d/2}}{\Gamma(d/2)}$ and therefore $\mathfrak d\_{2,d} = d$;
2. $p=1$: in this case $\mathrm{vol}\_d(B\_1^d) = \frac{2^d}{d!}$ and $\mathrm{vol}\_{d-1}(\partial B\_1^d) = \frac{2^d \sqrt{d}}{(d-1)!}$, and therefore $\mathfrak d\_{1,d} = d\sqrt{d}$
3. $p=\infty$: in this case $\mathrm{vol}\_d(B\_\infty^d) = 2^d$ and $\mathrm{vol}\_{d-1}(\partial B\_\infty^d) = 2^d n$. Therefore $\mathfrak d\_{\infty,d} = d$.


From the above examples, my (very wild) guess is that $\mathfrak d\_{p,d} \asymp d^{1+1/p-1/2}$ for $1\leq p\leq 2$ and $\mathfrak d\_{p,d} \asymp d$ for $p\geq 2$. But of course I could be very wrong. The challenge for general $p$ is the apparent difficulty in evaluating the volume of $L\_p$ sphere areas (e.g., [Surface area of an $\ell\_p$ unit ball?](https://mathoverflow.net/questions/234314/surface-area-of-an-ell-p-unit-ball)), but I'm hoping that the (asymptotic) ratio $\mathfrak d\_{p,d}$ for general $p$ is potentially easier to evaluate.


**Edit:** I realized that, because the volume of a unit $\ell\_p$ *ball* does have closed forms for general $p\in[1,\infty]$, the question can be answered if we know the **asymptotic dependency of dimension $d$ of the (unit) $\ell\_p$-surface area**. This question was explicitly mentioned in the comment under the accepted answer of the following question
[Surface area of superellipsoid (dice)](https://mathoverflow.net/questions/80243/surface-area-of-superellipsoid-dice?rq=1)
Unfortunately, the OP in that question did not pursue this direction.



Volume ratio of general $\ell_p$ balls and surfaces
We have managed to obtain a complete solution to this problem, by applying the [divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem) and the classical results of [Naor and Romik](http://www.numdam.org/article/AIHPB_2003__39_2_241_0.pdf) on the affinity between cone and surface measures in $\ell\_p$ balls. 


The results are different than what I have conjectured in the problem statement. In particular, it is proved that **$\mathfrak d\_{p,d} \asymp d^{1/2+1/p}$ for all fixed $p<\infty$ and $d\to\infty$.** Therefore, there is a phase transition between $p\to\infty$ and $p=\infty$. Such phase transition is because of the asymptotic regime we're considering ($p$ fixed, and $d$ goes to infinity afterwards). In cases where both $p$ and $d$ go to infinity simultaneously, the question remains unsolved.


The proof idea is to first use the **divergence theorem** to obtain
$$
d\times \mathrm{vol}\_d(\mathbb B\_p^d) = \mathrm{vol}\_{d-1}(\partial\mathbb B\_p^d)\times \int\_{\partial\mathbb B\_p^d}\frac{\mathrm{d}\sigma\_p^{d-1}(v)}{\sqrt{\sum\_{i=1}^d|v\_i|^{2p-1}}}.
$$
Here $\sigma\_p^{d-1}(v)$ is the **surface measure** on $\partial\mathbb B\_p^{d-1}$, meaning that $\sigma\_p^{d-1}(A)=\mathrm{vol}\_{d-1}(A)/\mathrm{vol}\_{d-1}(\partial\mathbb B\_p^d)$ for all measurable $A\subseteq\mathbb B\_p^{d-1}$.
The work of [Naor and Romik](http://www.numdam.org/article/AIHPB_2003__39_2_241_0.pdf) showed that this surface measure is very close (in TV distance) to a **cone measure** $\gamma\_p^{d-1}$, which is equivalently the distribution of $X=Z/\|Z\|\_p$ where $Z=(Z\_1,\cdots,Z\_d)$ consists of i.i.d. random variables distributed according to a generalized Normal distribution $p(\cdot)\propto \exp\{-|\cdot|^p\}$. It is then possible to use Chebyshev's inequality to get an adequate estimate of the integral term above, which yields an asymptotic formula for $\mathfrak d\_{p,d}$.


