an Alexandrov space
/ al-ek-SAHN-drof /
Think of the surface of a convex body — a sphere, a cone, the boundary of a cube. Such surfaces are 'positively curved' in a robust way: triangles drawn on them are fatter than flat triangles, and corners or ridges only make them fatter still, never thinner. An Alexandrov space packages exactly this intuition into a purely metric definition. It is a length space whose curvature is bounded below by some number kappa, captured entirely through how triangles compare to model triangles, with no smoothness required.
The definition uses comparison triangles. Fix kappa. A complete length space X has curvature bounded below by kappa if, for every small geodesic triangle in X, the triangle is at least as fat as its comparison triangle in the model space M_kappa: for any two points x, y on the sides, d(x, y) >= |x' - y'| where x', y' are the corresponding points in the comparison triangle. Equivalently, the angle at each vertex of the real triangle is at least as large as the corresponding model angle (the so-called Toponogov property). An Alexandrov space (with curvature bounded below) is such a space; one usually also asks it to be locally compact and geodesic. Cones, convex surfaces, quotients of manifolds by isometric group actions, and crucially Gromov-Hausdorff limits of manifolds with a uniform lower sectional curvature bound all qualify.
Alexandrov spaces are the natural home for limits in comparison geometry: the class of Riemannian manifolds with sectional curvature >= kappa is not closed, but its Gromov-Hausdorff closure lands you exactly among Alexandrov spaces with curvature >= kappa, which is why they appear whenever you take limits or collapse. They have a remarkable amount of structure for objects defined by an inequality — a well-defined Hausdorff dimension (an integer), tangent cones at every point, a notion of angle, and a dense smooth-manifold part — yet they can have genuine singularities like cone points. A caution: 'curvature bounded below' (Alexandrov) and 'curvature bounded above' (CAT(kappa)) are opposite worlds with opposite inequalities and very different singular behavior; do not conflate them. A space can satisfy both only in special, essentially flat-like cases.
A flat cone, obtained by rolling up a sector of the plane of angle less than 2pi, is an Alexandrov space with curvature bounded below by 0. Away from the tip it is locally flat, but at the cone point the total angle is deficient (less than 2pi), which makes triangles enclosing the tip strictly fatter than Euclidean ones — positive curvature concentrated at a single singular point. By contrast, gluing in a sector of angle more than 2pi gives a saddle-like cone with an angle excess, which is NOT an Alexandrov space of curvature >= 0, since such triangles are thinner than flat.
A flat cone: locally flat, but the angle deficit at the tip gives curvature bounded below by 0.
Curvature bounded below does not mean smooth: an Alexandrov space can have cone points, edges, and lower-dimensional strata. Its Hausdorff dimension is an integer, but the singular set is generally nonempty and can be subtle.