This is another proof which rests on the explicit formula for the volume $c_p(n)$ of the n-dimensional $l^p$ ball $B_p(n)=\{x \in \mathbb R^n: \sum_{i=1}^n |x_i|^p \leq 1\}$, namely $$c_p(n)=\frac{\left (\frac 2p \Gamma (1/p) \right )^n}{\Gamma (1+n/p)}.$$ If $\phi(x)=\sum_{i=1}^n |x_i|^p$, then $|\nabla \phi|^2 =p^2 \sum_{i=1}^n |x_i|^{2(p-1)}$ and using the coarea formula $$ r^{n/p} c_p(n)=m_n\{\phi \leq r\}=\int_0^r dt \int_{\{\phi=t\}}|\nabla \phi|^{-1}\, dt. $$ Differentiating with respect to $r$ and setting $r=1$ we get $$\frac np c_p(n)= \int_{\{\phi=1\}}|\nabla \phi|^{-1}\, dt \geq \frac{1}{p \sqrt n} m_{n-1}\{\phi=1\} $$ so that $m_{n-1}\{\phi=1\} \leq n^{\frac 32}c_p(n) \to 0$, as $n \to \infty$, by Stirling's formula for the $\Gamma$ function.

Note that when $p \geq 2$ then $2p-2 \geq p$ and $|\nabla \phi|^2 \leq p^2$ on $\{\phi=1\}$ so that the exponent $3/2$ improves to 1. When $p <2$ a better exponent follows from Holder.

This is another proof which rests on the explicit formula for the volume $c_p(n)$ of the n-dimensional $l^p$ ball $B_p(n)=\{x \in \mathbb R^n: \sum_{i=1}^n |x_i|^p \leq 1\}$, namely $$c_p(n)=\frac{\left (\frac 2p \Gamma (1/p) \right )^n}{\Gamma (1+n/p)}.$$ If $\phi(x)=\sum_{i=1}^n |x_i|^p$, then $|\nabla \phi|^2 =p^2 \sum_{i=1}^n |x_i|^{2(p-1)}$ and using the coarea formula $$ r^{n/p} c_p(n)=m_n\{\phi \leq r\}=\int_0^r dt \int_{\{\phi=t\}}|\nabla \phi|^{-1}\, d\sigma_t. $$ Differentiating with respect to $r$ and setting $r=1$ we get $$\frac np c_p(n)= \int_{\{\phi=1\}}|\nabla \phi|^{-1}\, d\sigma_1 \geq \frac{1}{p \sqrt n} m_{n-1}\{\phi=1\} $$ so that $m_{n-1}\{\phi=1\} \leq n^{\frac 32}c_p(n) \to 0$, as $n \to \infty$, by Stirling's formula for the $\Gamma$ function.

Note that when $p \geq 2$ then $2p-2 \geq p$ and $|\nabla \phi|^2 \leq p^2$ on $\{\phi=1\}$ so that the exponent $3/2$ improves to 1. When $p <2$ a better exponent follows from Holder.