OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support \mathop{RCD}\nolimits ((n-1)\kappa,n) condition with reference measure \mathcal H^n and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.
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MANUSCRIPT 01
Gigli’s distributional curvature characterization of Alexandrov spaces
For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure \mathcal H^n, it is a full-support RCD((n-1)\kappa,n) space whose distributional sectional curvature is at least κ in Gigli's original global test…
Full source abstract
For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure \mathcal H^n, it is a full-support RCD((n-1)\kappa,n) space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.
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Weak Hessian bounds along every geodesic in RCD spaces
On a full-support RCD(K,N) space with 1< N< ∞, we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.
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