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

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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.


  1. Robert A. Adams and John J. F. Foutnier, Sobolev Space (2nd Edition, 2003), p82 ↩︎

  2. Lawrence C. Evans, Partial Differential Equations (2nd Edition, 2010), p713 ↩︎