Q:

Visual understanding of convergence of domains in the sense of Fisher

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to explain to me what this second property actually means?
The definition states: A sequence of domains $(\Lambda_m)$ converges to $\mathbb R^m$ in the sense of Fisher if 

$\lim_{m\to\infty}|\Lambda_m|=\infty$
as $\epsilon\to0$, 
$$\sup_m \frac{|\partial_{\epsilon \operatorname{diam}\Lambda_m } \Lambda_m|}{|\Lambda_m|}\to 0$$
where $\partial_r \Lambda = \{x:\operatorname{dist}(x,\partial \Lambda)\le r$.

A:

Let's take $m=2$, where visualization is easier. 
The first bullet item simply says that the area of domains grows to infinity. This by itself is a pretty weak condition, e.g., rectangles of size $m^2 \times (1/m)$ satisfy it without being a geometrically satisfactory approximation to the plane. 
Informally, the second condition forces the domains to maintain bounded aspect ratio by requiring their perimeter  to not be too large compared to the area. It's not stated in terms of perimeter itself (which for some domains is difficult to define) but in terms of closely related quantity; the area of $r$-neighborhood of the boundary. Specifically, $r=\epsilon \operatorname{diam}\Lambda_m$, which means we look at the boundary at a scale smaller than the diameter of domain. 
Consider rectangles $a_m\times b_m$ with $a_m\ge b_m$. The diameter is roughly $a_m$. The area of $(\epsilon a_m)$-neighborhood of the boundary is roughly $\epsilon a_m^2$, by looking at the contributions of longest sides only. (All these estimates are up to multiplicative constants.) So, the requirement is 
$$
\sup_m \frac{\epsilon a_m^2}{a_m b_m} \to 0,\quad \epsilon\to0
$$
This is equivalent to the aspect ratio $a_m/b_m$ being bounded -- which, I hope, is a geometrically intuitive condition. 
So, neither $m^2\times (1/m)$ nor $m\times 1$ nor $m\times \sqrt{m}$ rectangles converge to the plane in the sense of Fisher. But $m\times m$ and $(10m)\times m$ rectangles do.

