What is the terminology for this process: $(A,\epsilon)\mapsto\{x\in A:N_\epsilon(x)\subseteq A\}$?

Question

Let $(X,d)$ be a metric space and $A\subseteq X$, and $N_\epsilon(x):=\{y\in X:d(x,y)<\epsilon\}$.

Fix $\varepsilon>0$. Consider the set

$$ \{x\in A:N_\varepsilon(x)\subseteq A\} $$

Intuitively, this is like a padded interior of $A$. It is the set of points in $A$ which are not too close to the boundary. For example if $A$ is a ball of radius $r$ then the above set is a ball of radius $r-\epsilon$. If $A$ is an interval $[a,b]$ then the above set is $[a+\epsilon,b-\epsilon]$. Is there a standard name for this type of set?

Note: this is not the same as an open set. An open set says there exists such an epsilon for every point, but I have already fixed my epsilon from the start.

Answer 1

Define $A_\epsilon := \{x \in A \mid N_\epsilon(x) \subseteq A\}$. My understanding is that you wish to put a name on $A_\epsilon$.

There is likely to be no specific name for this specific object.

Let's try to do something better, namely, link $A_\epsilon$ to something already known and studied in the literature. A good candidate is $((A')^\epsilon)'$, where $A' := X\setminus A$ is the set-compliment of $A$ relative to $X$ and $B^\epsilon := \{x \in X \mid d(x,B) < \epsilon\}$ is the $\epsilon$-enlargement of $B \subseteq X$, with $d(x,B) := \underset{b \in B}{\inf}\; d(x,b)$ is the distance of $x$ from $B$. $\epsilon$-enlargements are well-studied in geometric probability theory. For example,

• The Prokhorov metric is defined via such objects.
• Strassen's marriage lemma can be formulated via such objects.
• (Geometric theory of) concentration of measure phenomenon is built on subject objects
• etc.

Claim. $A_\epsilon = ((A')^\epsilon)'$.

Proof. It suffices to show that $(A_\epsilon)' = (A')^\epsilon$. By direct computation, $$ \begin{split} x \in (A_\epsilon)' \iff N_\epsilon(x) \not\subseteq A \iff \exists y \in A' \mid d(x,y) < \epsilon &\iff d(x,A') < \epsilon\\ &\iff x \in (A')^\epsilon. \end{split} $$