Uniform domains close to a ball

Each pair of points in a uniform domain $U$ is joined by a “cigar”, the image of a curvilinear cone under a Möbius transformation. We obtain a few geometric properties of such domains under the condition that the angles at the vertices of the “cigars” are close to $\pi$. We prove that if $\overline{\mathbb R^n}\setminus\overline U=U^*\ne\varnothing$ then $U^*$ is uniform too. If $\partial U$ is unbounded then it is almost flat, i.e., for every ball $B(x,r)$, its intersection with $\partial U$ lies in the $\delta r$-neighborhood of some hyperplane. These properties are possessed by the images of balls under quasiconformal mappings close to conformal mappings.