Two Kneser--Poulsen-type Inequalities in Planes of Constant Curvature

TL;DR

This paper extends Kneser--Poulsen-type inequalities to hyperbolic and Euclidean planes, showing that certain geometric measures like perimeter and area behave monotonically under specific rearrangements of disks.

Contribution

It generalizes known Euclidean inequalities to hyperbolic and spherical geometries using a novel adaptation of existing methods.

Findings

Perimeter of convex hulls of disks does not increase under rearrangements.

Area of intersection of disks in hyperbolic plane does not decrease after rearrangements.

Results unify and extend inequalities across different constant curvature planes.

Abstract

We show that the perimeter of the convex hull of finitely many disks lying in the hyperbolic or Euclidean plane, or in a hemisphere does not increase when the disks are rearranged so that the distances between their centers do not increase. This generalizes the theorem on the monotonicity of the perimeter of the convex hull of a finite set under contractions, proved in the Euclidean plane by V. N. Sudakov, R. Alexander, V. Capoyleas and J. Pach. We also prove that the area of the intersection of finitely many disks in the hyperbolic plane does not decrease after such a contractive rearrangement. The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis.