Uniform Cone Condition with Unit Normal Vector

Question

I was going through this article from M. BRAMSON, K. BURDZY AND W. KENDALL https://projecteuclid.org/download/pdfview_1/euclid.aop/1362750941 where the uniform interior cone condition is given this way:

A domain $D$ is said to satisfy a uniform interior cone condition, based on radius $\delta > 0$ and angle $\alpha \in (0,\frac{\pi}{2}]$, if, for every $x \in \partial D$, there is at least one unit vector $m$ such that the cone $C(m) = \{z : \langle z, m \rangle > |z| \cos(\alpha)\}$ satisfies $$ (y + C(m)) \cap B(x,\delta) \subset D, \qquad \forall y \in D \cap B(x,\delta). $$

Because of the terminology and of the geometric intuition, it seems to me that this definition should be equivalent to the following:

For all $x \in \partial D$, there exists a radius $h$ and an angle $\beta$ such that $x + su \in D$ for all $u \in \mathbb{S}^{n-1}, |u \cdot n_x| > \beta$, $s \in (0,h)$, where $n_x$ the inward normal vector at $x$.

I believe the second definition is basically the first one with the choice $m = n_x$ for all $x$, and that this choice can be made at all $x \in \partial D$ if the uniform interior cone condition in the sense of the first definition holds.

Am I missing something here ?

Answer 1

There are at least two reasons why the two definitions are different:

1. In your second definition, you use the unit normal vector, which is not defined if the domain boundary is not sufficiently smooth. In fact, bounded Lipschitz domains already satisfy a uniform cone condition.
2. Why we call it a uniform cone condition is that just one set of parameters $\alpha$ are $\delta$ should work everywhere. Take the planar domain $\{(x,y):x>1,\,0<y<1/x\}$. This would fulfill your second definition but would not satisfy a uniform cone condition.