Relating dispersion and discrepancy
Since balls have positive volume, there is a close relationship
between discrepancy, which is measurebased, and dispersion, which is
metricbased. For example, for any
,

(5.23) 
which means lowdiscrepancy implies lowdispersion. Note that the
converse is not true. An axisaligned grid yields high discrepancy
because of alignments with the boundaries of sets in
, but the
dispersion is very low. Even though lowdiscrepancy implies
lowdispersion, lower dispersion can usually be obtained by ignoring
discrepancy (this is one less constraint to worry about). Thus, a
tradeoff must be carefully considered in applications.
Steven M LaValle
20200814