We know about volume: The $L\_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of the inscribed $L\_p$ ball $\{x : \|x\|\_p \leq \tfrac{1}{2}\}$ goes to zero as the dimension $d \to \infty$. (For example, a simple generalization of [this argument](https://mathoverflow.net/a/8270/29697).)

My question is the analogous one for surface area. The surface area of the $L\_2$ ball goes to zero as the dimension diverges. But the surface area of the $L\_{\infty}$ ball is $2d \to \infty$. So, we might wonder if there is a "happy medium" $p \in (2,\infty)$ where the surface area is constant in all dimensions. But I doubt it. **Does the surface area of the radius-$\tfrac{1}{2}$ $L\_p$ ball go to zero for all $p < \infty$**?

Asymptotics about the surface area of Lp balls are [apparently hard to pin down](https://mathoverflow.net/questions/234314/surface-area-of-an-ell-p-unit-ball). Tools from [this mathoverflow Q&A](https://mathoverflow.net/questions/293840/volume-ratio-of-general-ell-p-balls-and-surfaces) and linked paper might be useful.