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An invitation to Alexandrov geometry: CAT(0) spaces

Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.

preprint2026arXivOpen access
Stephanie AlexanderVitali KapovitchAnton Petrunin
math.ATmath.MGmath.DG
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Stephanie AlexanderVitali KapovitchAnton Petrunin

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