Differential Inequalities for Distance Comparison

2017/01/17 by Limoncu, Murat, Koçak, Şahin
#51F99 #52A55 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1701.04725

Abstract

Comparison of 1-dimensional distance functions is a basic tool in Alexandrov geometry and it is used to characterize spaces with curvature bounded above or below. For the zero curvature bound there is a differential inequality which enables one to check this comparison directly on a given smooth 1-dimensional distance function. In this note we give a generalization of this property to arbitrary curvature bounds.

Related

• A rigidity theorem in Alexandrov spaces with lower curvature bound
• Pythagorean Theorem & curvature with lower or upper bound
• Isoperimetric characterization of upper curvature bounds
• One-Sided Derivative of Distance to a Compact Set
• Sharp Hardy inequalities involving distance functions from submanifolds of Riemannian manifolds