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I wonder if the following definitions of the Gaussian surface measure are equivalent.

First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the $\ell_p$ sense as \begin{align} A^{\epsilon}_p = \{ y \in \mathbb{R}^n \colon \text{there exists}~ x \in A ~\text{such that}~ \| x - y \|_{p} \leq \epsilon \}, \end{align} where $1\leq p\ \leq + \infty$. We denote $\gamma$ as the standard $n$-dimensional Gaussian measure $N(0, I_n)$, then the Gaussian surface measure is defined as \begin{align} \tau (A) = \lim_{\epsilon \rightarrow 0} \frac{ \gamma( A^{\epsilon}_p \setminus A) }{ \epsilon}. \end{align}

Second, another definition of Gaussian surface measure is \begin{align} \tau (A) = \int _{\partial A} \varphi(x) \mathrm{d}\sigma(x), \end{align} where $\varphi$ is the density of $N(0, I_n)$ and $\mathrm{d}\sigma(x)$ is the standard surface measure in $\mathbb{R}^n$.

In ``On the maximal perimeter of a convex set in R n with respect to a Gaussian measure'' it is shown that these two notions are equivalent for $p =2$, that is, the neighbors are considered in the sense of $\ell_2$-balls.

I was wondering whether such an equivalence holds for all $p\in [1, +\infty]$ in general? If not, for what $p$ and what set $A$ can we establish such an equivalence?

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No: unless $p = 2$, these definitions are not equivalent. The value that you define in the question for a rectangle $A$ whose sides are parallel to $y = \pm x$ is easily found to be $2^{1/2-1/p} \int_{\partial A} \varphi(x) d\sigma(x)$.

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