Two Kneser–Poulsen-type inequalities in planes of constant curvature

Abstract

Dedicated to Ted Bisztriczky, Gábor Fejes Tóth, and Endre Makai on the occasion of their 70th birthdays.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 [8], R. Alexander [1], V. Capoyleas and J. Pach [3]. 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 [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4].The following consequence of the Euclidean Kneser-Poulsen conjecture was proved by V. N. Sudakov [8], R. Alexander [1], V. Capoyleas and J. Pach [3].

On the intrinsic volumes of intersections of congruent balls

“…On the other hand, the author and R. Connelly [3] proved (3) as well as (4) for N not necessarily congruent circular disks and for all N > 1 in E 2 . Very recently B. Csikós and M. Horváth [8] (resp., I. Gorbovickis [13]) gave a positive answer to the Gromov-Klee-Wagon problem in H 2 (resp., S 2 for circular disks having radii at most π 2 ). However, both (3) and ( 4) remain open in E d for all d ≥ 3.…”