The central set and its application to the Kneser-Poulsen conjecture

TL;DR

This paper introduces new results on central sets in Riemannian manifolds and applies them to prove special cases of the Kneser-Poulsen conjecture in 2D spherical and hyperbolic spaces.

Contribution

It provides novel insights into central sets and extends the Kneser-Poulsen conjecture to new cases in non-Euclidean geometries.

Findings

Proved new special cases of the Kneser-Poulsen conjecture in 2D sphere and hyperbolic plane.

Developed new results about central sets in Riemannian manifolds.

Extended understanding of volume behavior under ball rearrangements.

Abstract

The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the two-dimensional sphere and the hyperbolic plane.