---
tag: 71AWVBG6KH
kind: concept
name: Alexandrov space
msc: [53C23, 53C20]
specializes: [0VKD1TP7F3]
sources:
  - url: https://en.wikipedia.org/wiki/Alexandrov_space
same_as:
  wikidata: Q28401130
  wikipedia: 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|GT56RTG9X5]].
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.

## Explanations

### L1 human

In a space of curvature at least $0$, triangles are at least as fat as flat triangles with the same side lengths.
The surface of a convex body, like a cube or an egg, is a good picture: it has corners, but it curves outward everywhere.
The surface of a cube is an Alexandrov space of curvature at least $0$, although it is not smooth at the corners.
A saddle surface is not, because its triangles are thinner than flat ones.