Convexity of the Urysohn universal space (Q2822000)

The authors note that \textit{P. Urysohn} in [Bull. Sci. Math., II. Sér. 51, 43--64 and 74--90 (1927; JFM 53.0556.01)] constructed a complete separable metric space \(\mathbb U\), referred to as \textit{the Urysohn universal space}, that contains an isometric copy of every complete separable metric space and has the \textit{finite transivity property}, which means that every isometry between the finite subsets of \(\mathbb U\) extends to an isometry of \(\mathbb U\) onto itself. They outline Urysohn's construction of \(\mathbb U\) and its metric \(\rho\) and study various convexity properties of \((\mathbb U,\rho)\). The terminology used, where \(B(x,r)\) denotes the closed ball centered at \(x\) with radius \(r\), is as follows. A metric space \((X,d)\) is said to be \textit{hyperconvex} if \(\cap_{i\in I}B(x_i,r_i)\neq\emptyset\) for every collection of balls \(B(x_i,r_i)\) in \(X\) for which \(d(x_i,x_j)\leq r_i+r_j\) for all \(i,\;j\in I\). If the preceding holds for every finite set \(I\) then \((X,d)\) is said to satisfy \textit{the finite ball intersection property}. Two of the results the authors obtain are the following: The Urysohn universal space \((\mathbb U,\rho)\) satisfies the finite ball intersection property, but it is not hyperconvex.