Shrinking Starlike Sets

Abstract

‘Shrinking Starlike Sets’ presents a proof that null decompositions with recursively starlike-equivalent elements shrink. Unlike in previous chapters, the shrink is produced abstractly rather than explicitly. The chapter proceeds in steps, first establishing the shrinking of null decompositions with starlike elements, then of those with starlike-equivalent elements, and then finally of those with recursively starlike-equivalent elements. In the eventual proof of the disc embedding theorem, a decomposition of the 4-ball consisting of recursively starlike-equivalent elements will be produced. These consist of the ‘holes’, along with certain ‘red blood cell discs’. The result from this chapter will be used to show that the decomposition shrinks; this is called the α-shrink.