TITLE: Gaussian Surface Area of Positive Semidefinite Cone
QUESTION [7 upvotes]: Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the $\ell_2$ sense as
\begin{align}
A^{\epsilon} = \{ y \in \mathbb{R}^n \colon \text{there exists}~ x \in A ~\text{such that}~ \| x - y \|_{2} \leq \epsilon \}.
\end{align}
 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} \setminus A) }{  \epsilon}.
\end{align}
By results of Keith Ball (the reverse isoperimetric problem for Gaussian measure https://link.springer.com/article/10.1007/BF02573986), there is a universal upper bound of the Gaussian surface area of any convex set. 
My questions are: 1) when is Keith Ball's upper bound tight? 2) what is a tight upper bound of the Gaussian surface area of the cone of positive semidefinite matrices?

REPLY [3 votes]: Regarding the tightness of Ball's bound, which is $O(n^{1/4})$ for a convex set in dimension $n$, this was proven to be sharp (up to a constant factor) by Nazarov, in 
Nazarov, Fedor, On the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to a Gaussian measure, Milman, V. D. (ed.) et al., Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 2001--2002. Berlin: Springer (ISBN 3-540-00485-8/pbk). Lect. Notes Math. 1807, 169-187 (2003). ZBL1036.52014.