📂MetricSpace

Sets Outside/Inside a Certain Distance from the Boundary of a Set

Definition

Let us assume an open set $\Omega \subset \mathbb{R}^n$ is given. Then, $\Omega_{<\delta}$ and $\Omega_{>\delta}$ are defined as follows.

$$ \begin{align*} \Omega_{<\delta} :=& \left\{ x\in\Omega : \mathrm{dist}(x, \mathrm{bdry}\Omega)<\delta \right\} \\ \Omega_{>\delta} :=& \left\{ x\in\Omega : \mathrm{dist}(x, \mathrm{bdry}\Omega)>\delta \right\} \end{align*} $$

Explanation

Such sets are usefully employed in partial differential equations, functional analysis, etc. Depending on the textbook, there are cases where it’s $\Omega_\delta=\Omega_{<\delta}$¹ and cases where it’s $\Omega_\delta=\Omega_{>\delta}$². In those instances, it’s best to faithfully follow the notation used in the class or textbook. Fresh Shrimp Sushi House uses both definitions, hence the notation was defined as above.