Quadratic metric comparisons
Nina Lebedeva, Anton Petrunin, Vladimir Zolotov
TL;DR
The paper investigates how quadratic distance inequalities constrain the global geometry of length spaces by encoding inequalities as a cone K in W_n via associated forms. It shows a globalization principle: for a nontrivial 4-point quadratic condition, local validity implies global validity and yields Alexandrov spaces with nonnegative curvature; a single negative-type 4-point inequality already enforces full curvature-type behavior. The results connect rank-one, negative-type inequalities to concrete geometric embeddings (e.g., into r·S^1×ℝ^3 or a tripod product) and establish a Toponogov-like globalization phenomenon for 4-point comparisons. The work builds a framework linking 4-point metric data, associated forms, ultralimit stability, and Alexandrov curvature through a rigorous local-to-global analysis.
Abstract
We study the effects on length spaces imposed by quadratic inequalities on the six distances between the points in every quadruple.