How to Net a Lot with Little : Small " - Nets for

It is known that in general range spaces of VC-dimension d > 1 require "-nets to be of size at least (d " log 1 "). We address the question whether this general lower bound is valid for the special range spaces that typically arise in computational geometry. We show that disks and pseudo-disks in the plane as well as halfspaces in R 3 allow "-nets of size only O(1 "), which is best possible up to a multiplicative constant. The analogous questions for higher-dimensional spaces remain open.