Uniform domains and uniform domain decomposition property in real normed vector spaces (Q2857597)

A uniform domain \(D\) in \(\mathbb{R}^n\) has the property that a pair of points lie in a simpler domain (quasiball, bi-lipschitz ball) inside \(D\) [\textit{O. Martio} and \textit{J. Sarvas}, Ann. Acad. Sci. Fenn., Ser. A I, Math. 4, 383--401 (1979; Zbl 0406.30013); \textit{G. J. Martin}, Trans. Am. Math. Soc. 292, 169--191 (1985; Zbl 0584.30020)]. This property can also be used to characterize uniform domains. The method to prove this type of results is based on the use of quasihyperbolic metric and its geodesics. In a domain \(D\) in an infinite dimensional vector space \(V\) there need not be such a geodesic connecting two points in \(D\), and hence this method does not work there. Using a direct construction based on a chain of balls the authors show that a uniform domain \(D\) in a real vector space can be characterized by the following property: For each pair of points \(x_1, \, x_2 \in D\) there exists a subdomain \(D_1\) such that \(x_1, \, x_2 \in D_1\) and \(D_1\) is a simply connected \(c\)-uniform domain.