Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ have the smallest "boundary" content (aka surface areas). More formally

[Gaussian Isoperimetry] If $A \subseteq \mathbb R^n$ is Borel and $H$ is a half-space with $\gamma_n(A) = \gamma_n(H)$, then $$ \gamma_n(A^\epsilon) \ge \gamma_n(H^\epsilon),\;\forall \epsilon > 0. $$

where

• $A^\epsilon := \{x \in \mathbb R^n | d(x,A) \le \epsilon\}$ is the $\epsilon$-blowup of $A$, and

• $d(x,A):=\inf_{x' \in \mathbb R^n}\|x'-x\|_2$ is the distance of $x$ from $A$.

Question

In the above result, if the norm $p$ is changed from $\ell_2$ to some general $\ell_p$ with $p \in [1, \infty]$, the minimal boundary shapes $H$ change from half-space to what ?

N.B.: I'm particularly interested in the cases $p = 1$ and $p=\infty$.