A globalization for non-complete but geodesic spaces (Q330851)

The article shows that given a geodesic space \(X\) with the property that for any point \(x \in X\) there exists a neighborhood \(\Omega \ni x\) such that the \(\kappa\)-comparison holds for any quadruple of points in \(\Omega\), then the completion of \(X\) is an Alexandrov space with curvature \(\geq \kappa\). This answers a question asked by Viktor Schroeder around 2009. Here an Alexandrov space with curvature \(\geq \kappa\) is a complete length space such that for any quadruple of points \((p; x^1; x^2; x^3)\) the \((1+3)\)-point comparison holds: \[ \measuredangle^\kappa(p^{x^1}_{x^2}) + \measuredangle^\kappa(p^{x^2}_{x^3}) + \measuredangle^\kappa(p^{x^3}_{x^1}) \leq 2 \pi. \]