How to net a lot with little: small ε-nets for disks and halfspaces

It is known that in general range spaces of VC-dimension <italic>d</italic> > 1 require <italic>ε</italic>-nets to be of size at least &OHgr;(<italic>d</italic>/<italic>ε</italic> log 1/<italic>ε</italic>). We investigate 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<supscrpt>3</supscrpt> allow <italic>ε</italic>-nets of size only <italic>&Ogr;</italic>(1/<italic>ε</italic>), which is best possible up to a multiplicative constant. The analogous questions for higher-dimensional spaces remain open.

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