Sufficient conditions of $\alpha$-accessibility of domain in nonsmooth case

This paper continues the study of $\alpha$-accessible domains in $\mathbb{R}^n$ in nonsmooth case. A domain $\Omega\subset \mathbb{R}^n, 0\in\Omega$ is $\alpha$-accessible (see [2], [3]), $\alpha\in [0;1)$; if for every point $p\in\partial\Omega$; there exists a number $r=r(p)>0$ such that the cone $K _{+}(p,\alpha,r)=\{x\in \mathbb{\bar{R}}^{n}(p,r) : (x-p,\frac{p}{||p||})\ge ||x-p||\cos\frac{\alpha\pi}{2}\}$ is included in $\Omega'=\mathbb{R}^n\setminus\Omega$. $\alpha$-accessible domains are starlike domains with respect to the inner point zero and satisfy cone condition which is important for applications,such as the theory of integral representations of functions, imbedding theorems, the questions of the boundary behavior of functions, the solvability of Dirichlet problem. Appropriate conditions of $\alpha$-accessibility of domain, defined by the inequality $F(x)<0$ for a continuous function $F$ in $\mathbb{R}^n$ are obtained in [1]. There were obtained and some sufficient conditions of $\alpha$-accessibility. In this article, these sufficient conditions have been significantly strengthened. Here is an example of their use. Conditions, obtained in the theorem, and consequences to it are also sufficient for the starlikeness of the set (the case when $\alpha=0$). In the smooth case the criterion for the $\alpha$-accessibility of domain was obtained in [2].