Weak Cone Condition

Definition¹

Let $\Omega \subset \mathbb{R}^{n}$ be called an open set. Suppose a point $x \in \Omega$ is given. Let $R(x)$ be the set of all $y$ that ensure the line segment from $x$ to $y \in \Omega$ is included within $\Omega$ again. That is, $R(x)$ is the set of points on all lines starting from $x$ within $\Omega$. And let $\Gamma (x)$ be defined as follows.

$$\begin{align*} \Gamma (x) :=&\ \left\{ y \in R(x) : |y-x| \lt 1\right\} \\ =&\ R(x) \cap B(x,1) \end{align*}$$

If there exists a $\delta \gt 0$ that satisfies the following condition, $\Omega$ is said to satisfy the weak cone condition.

$$\mu_{n} \Big( \Gamma (x) \Big) \ge \delta, \quad \forall\ x \in \Omega$$

Here, $\mu_{n}$ is the Lebesgue measure.