## 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
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. 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
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under arbitrary contractions. In Section 8.7 we extend the result of Section 8.4 to all intrinsic volumes of r-ball bodies in
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. In Section 8.8 we apply this result to prove the Kneser-Poulsen Conjecture for intersections of sufficiently many equal balls in
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under uniform contractions. In Section 8.9 we prove the dual of this result for unions of sufficiently many equal balls in
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under uniform contractions. Finally, in Section 8.10 we present the proof of the Kneser-Poulsen Conjecture for unions (resp., intersections) of disks in
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or ℍ^{2}, under the condition that the first set has a simply connected interior.