1.5. WHAT IS AHEAD 17In the remainder of Part 1 (Chapters 2 to 7) we will consider properties in each
of these categories. A number of them can be used to characterize Euclidean disks
or half-planes. We will indicate when this is the case.
In Part 2 (Chapters 8 to 11) we will present proofs for some of the character-
izations mentioned above. Many of the arguments follow a series of implications.
There are four main series of implications, as well as some additional equivalences
proved. Some proofs not in the main series of implications belong naturally to the
discussion of the results and are presented in Part l.