Is there any equivalence between standard d dimensional Gaussian surface measure and d dimensional Hausdorff measure on boundary of convex sets?

Question

I am currently going through the papers of Nazarov (2003): "On the maximal perimeter of a convex set in $\Bbb R^n$ with respect to a Gaussian measure" (MR2083397, Zbl 1036.52014) and Ball (1993): "The reverse isoperimetric problem for Gaussian measure" (MR1243336, Zbl 0788.52010). To define the Gaussian perimeter of a convex set $C\subset \Bbb R^d$, Ball(1993) defined the integral over boundary $\partial{C}$ of $C$ with respect to $d-1$ dimensional Hausdorff measure as follows: $$\int_{\partial{C}}\phi_d(\mathbf{z})\mathcal{H}^{d-1}(d\mathbf{z}),$$ where $\phi_d(\mathbf{z})=(2\pi)^{-d/2}\exp\{-(\mathbf{z}^\prime \mathbf{z})/2\}$ is the density function of $d$ dimensional standard Gaussian random vector and $\mathcal{H}^{d-1}$ denote $(d-1)$ dimensional Hausdorff measure.

On the other hand, Nazarov (2003) defined the same quantity as follows: $$\int_{\partial{C}}\phi_d(\mathbf{z})d\sigma(\mathbf{z}),$$ where, $d\sigma(\mathbf{z})$ is the standard surface measure in $\Bbb R^d$.

My questions are as follows:

1. How can we prove the equivalence between these two definitions? By that I mean "is there any result that says for boundary of convex sets on $R^d$ we can actually interchange between surface measure and Hausdorff measure with appropriate dimension"? Any help or reference regarding this will be sincerely appreciated.

2. Suppose I believe there's some equivalence between these two measures. But can anyone please intuitively explain why Ball (1993) considered the dimension $(d-1)$ for Hausdorff measure ? Any result or reference will be helpful.

P.S.: I am new to these concepts. But I was just going through the book: Measure Theory and Fine Properties of Functions by Evans and Gariepy (MR3409135, Zbl 1310.28001).

1. They have mentioned that "if $k$ is an integer then $\mathcal{H}^k$ agrees with ordinary $k$-dimensional surface area on nice sets" (ref. page 82). What do they mean by that nice sets?

2. Another result (not sure of the reference!) says that "For a $k$-dimensional manifold $M\subset R^d$ of class $\mathcal{C}^1$ with $1\leq k\leq d$, M has Hausdorff dimension $d$ and that $\mathcal{H}^d(M)$ is the standard surface measure of $M$".

3. "The boundary of an $d$-manifold with boundary is an $(d-1)$-manifold."

Does the statements 1), 2) or 3) from this book help anyway to my questions 1) and 2)? Any help will be appreciated. Thanks in advance.

Answer 1

I would have answered basically (a) to avoid digging precise results myself.

"Nice" means that there are several classes of sets that have both "surface measure" (that can be defined as the Riemannian volume for submanifolds) and Hausdorff measure coincide, including $C^1$ submanifolds and boundaries of convex sets. The latter need not be $C^1$ (they can have edges and corners) so, as for other classes of sets one might be interested in, there is a question about how to define their "natural" measure, and several approaches are possible. Basically, if done right, they should all ultimately coincide; but some "irregular" sets have several existing definitions make sense without coinciding, hence the need to restrict.

About references, I would first look in Mattila's book Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. If you do not find the result there, it probably is in Federer's book Geometric Measure Theory, but it is very difficult to read (even statements need a lot of definition).

Answer 2

Generally, one uses the $(d - 1)$-Hausdorff measure for measuring the size of $\partial C$. If you wish to define "surface measure" by restricting the measure from the ambient space (or by local charts), then that requires some amount of regularity of the boundary. In your case, these two coincide. The boundary of a convex domain is locally the graph of a convex function, and convex functions are almost everywhere twice differentiable (theorem of Alexandrov). If you use the fact that for open sets in $\mathbb{R}^m$, the Lebesgue measure coincides with $\mathcal{H}^m$, you can prove the equivalence you want.

Everything I wrote above can be located in Evans-Gariepy. You can also look at Falconer's book.