Alexandrov space

Summary

An Alexandrov space is a metric space with a lower bound on its curvature, expressed by comparing triangles with triangles of a model surface. It generalizes a complete Riemannian manifold with sectional curvature bounded below. For fixed $k$ and a fixed bound on the dimension, the class of compact Alexandrov spaces is closed under Gromov–Hausdorff limits. This makes it the natural home of such limits. Some authors use the name also for spaces with curvature bounded above; here it means a lower bound.