I am trying to understand the relationship between what I call the inner Minkowski content of a set and its perimeter. I am looking for a class of sets that is large enough, but for which these concepts can be compared.
Suppose $A \subset \mathbb{R}^2$ is an open set of finite perimeter P(A). Let $A_r = \{x \in A : dist(x,\partial A)< r \}$, where $\partial A$ is the topological boundary of $A$. Note that $A_r$ lies inside of $A$.
Define the inner Minkowski content of $A$ by $$I(A):=\lim_{r \rightarrow 0}\frac{\mathcal{H}^2(A_r)}{r}.$$
Question 1: Suppose both the inner Minkowski content and the perimeter of $A$ are finite. Is it true that $$I(A)=P(A)?$$
If so, then:
Question 2: Is there a reasonably large collection of sets $\{A\}$ for which $I(A)=P(A)$ holds and the limit is uniform over all sets in the collection? (For example sets with Lipschitz boundary and uniformly bounded Lipschitz constant?)