The estimation of the length of a convex curve in two-dimensional Alexandrov space

TL;DR

This paper generalizes Toponogov's theorem to estimate the length of convex curves within two-dimensional Alexandrov spaces, extending classical Riemannian geometry results to more general metric spaces.

Contribution

It introduces a new length estimation theorem for convex curves in two-dimensional Alexandrov spaces, broadening the scope of geometric analysis beyond Riemannian manifolds.

Findings

Generalization of Toponogov's theorem to Alexandrov spaces

New length bounds for convex curves in Alexandrov spaces

Extension of classical geometric inequalities

Abstract

It is proved the generalization of Toponogov theorem about the length of the curve in two-dimensional Riemannian manifolds in the case of two-dimensional Alexandrov spaces.