Under Cone Conditions

Definition¹

Let $\Omega \subset \mathbb{R}^{n}$ be an open set. If there exists some finite cone such that for each $x \in \Omega$, there exists a finite cone $C_{x} \subset \Omega$ having $x$ as its vertex, then $\Omega$ satisfies the cone condition.

Explanation

For all $x\in \Omega$, if there exists $C_{x} \in \Omega$ as shown in the figure above, then $\Omega$ satisfies the cone condition. If $\Omega$ includes a pointed part as shown in the next figure, it can’t satisfy the cone condition.

$\Omega$, as shaped like a diamond pattern above, fails to meet the condition at least at 4 vertices. Regardless of the size of the cone, it’s clear that $C_{x}$ can never be included in $\Omega$.