I first give a definition
Let $\nu$ be a non-zero vector in $\mathbb R^2$, and for each $x\neq 0$ let $\angle(x,\nu)$ be the angle between the position vector $x$ and $\nu$. For given such $\nu, \kappa$ satisfying $0\leq\kappa\leq\pi$, the set
$$C=\{x\in\mathbb R^2: x=0,\angle(x,\nu)\leq \kappa/2 \}$$
is called a finite cone with axis direction $\nu$ and aperture angle $\kappa$ with vertex at the origin. Note that $x+C=\{x+y: y\in C\}$ is a finite cone with vertex at $x$ but the same dimensions and axis direction as $C$ and is obtained by parallel translation of $C$.
And the question is: Let $\Omega \in\mathbb R^2$ be unbounded domain but satifying bounded in one direction. So does $\Omega$ satisfy cone condition, and does $\mathbb R^2\setminus \bar{\Omega}$ contain semi-cone?
Thanks for answering so much!
Best Answer
Your problem is still underdetermined with its context staying unclear. Smooth boundary of unspecified smoothness usually implies a sufficiently smooth boundary when the context is clear, and not worse than Lipschitz in any case. A domain $\Omega$ is said to satisfy the cone condition if there is a finite cone $C$ of height $\rho$ and aperture $\kappa$ such that each $x\in\Omega$ is the vertex of some cone $C_x\subset\Omega$ congruent to $C$. Of course, an unbounded domain $\Omega$ bounded in one direction will satisfy the cone condition with some cone $C$ for sufficiently small $\rho$ and $\kappa$ in case $\partial\Omega$ is uniformly smooth. And of course, an open set $\mathbb{R}^2\backslash\overline{\Omega}$ will satisfy the cone condition with some cone $\widetilde{C}$. Can $\widetilde{C}$ be chosen as a semi-cone with respect to $C$? Not necessarily, if a semi-cone is to be understood directly as a finite cone of height $\rho/2$ and aperture $\kappa/2$. Indeed, a boundary $\partial\Omega$ may possess, say, certain narrow windings impenetrable for a cone $\widetilde{C}_x\,$. But the cone $C$ could as well be chosen with $\rho$ and $\kappa$ small enough for $\mathbb{R}^2\backslash\overline{\Omega}$ to fit the cone condition with the semi-cone $\widetilde{C}$.