ABSTRACT
In this chapter we present selected proofs of some theorems from Chapter 3 about the Kneser-Poulsen Conjecture. In Sections 8.1–8.3 we prove this conjecture, both for unions and for intersections of Euclidean disks in E 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq1762.tif"/> . In Section 8.4 we prove a Blaschke-Santaló-type inequality for r-ball bodies in spaces of constant curvature. In Section 8.5 we prove that the volume of the intersection of equal balls in a space of constant curvature increases under uniform contractions. In Section 8.6, we drop the condition of uniform contractions, and prove the Kneser-Poulsen Conjecture for unions and intersections of closed hemispheres in S d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq1763.tif"/> under arbitrary contractions. In Section 8.7 we extend the result of Section 8.4 to all intrinsic volumes of r-ball bodies in E d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq1764.tif"/> . In Section 8.8 we apply this result to prove the Kneser-Poulsen Conjecture for intersections of sufficiently many equal balls in E d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq1765.tif"/> under uniform contractions. In Section 8.9 we prove the dual of this result for unions of sufficiently many equal balls in E d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq1766.tif"/> under uniform contractions. Finally, in Section 8.10 we present the proof of the Kneser-Poulsen Conjecture for unions (resp., intersections) of disks in S 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq1767.tif"/> or ℍ2, under the condition that the first set has a simply connected interior.
